public static void e04nq( string start, E04..::..E04NQ_QPHX qphx, int m, int n, int ne, int nname, int lenc, int ncolh, int iobj, double objadd, string prob, double acol, int inda, int loca, double bl, double bu, double c, string names, int helast, int hs, double x, double pi, double rc, ref int ns, out int ninf, out double sinf, out double obj, E04..::..e04nqOptions options, out int ifail )
Public Shared Sub e04nq ( _ start As String, _ qphx As E04..::..E04NQ_QPHX, _ m As Integer, _ n As Integer, _ ne As Integer, _ nname As Integer, _ lenc As Integer, _ ncolh As Integer, _ iobj As Integer, _ objadd As Double, _ prob As String, _ acol As Double(), _ inda As Integer(), _ loca As Integer(), _ bl As Double(), _ bu As Double(), _ c As Double(), _ names As String(), _ helast As Integer(), _ hs As Integer(), _ x As Double(), _ pi As Double(), _ rc As Double(), _ ByRef ns As Integer, _ <OutAttribute> ByRef ninf As Integer, _ <OutAttribute> ByRef sinf As Double, _ <OutAttribute> ByRef obj As Double, _ options As E04..::..e04nqOptions, _ <OutAttribute> ByRef ifail As Integer _ )
public: static void e04nq( String^ start, E04..::..E04NQ_QPHX^ qphx, int m, int n, int ne, int nname, int lenc, int ncolh, int iobj, double objadd, String^ prob, array<double>^ acol, array<int>^ inda, array<int>^ loca, array<double>^ bl, array<double>^ bu, array<double>^ c, array<String^>^ names, array<int>^ helast, array<int>^ hs, array<double>^ x, array<double>^ pi, array<double>^ rc, int% ns, [OutAttribute] int% ninf, [OutAttribute] double% sinf, [OutAttribute] double% obj, E04..::..e04nqOptions^ options, [OutAttribute] int% ifail )
static member e04nq : start : string * qphx : E04..::..E04NQ_QPHX * m : int * n : int * ne : int * nname : int * lenc : int * ncolh : int * iobj : int * objadd : float * prob : string * acol : float * inda : int * loca : int * bl : float * bu : float * c : float * names : string * helast : int * hs : int * x : float * pi : float * rc : float * ns : int byref * ninf : int byref * sinf : float byref * obj : float byref * options : E04..::..e04nqOptions * ifail : int byref -> unit
- Type: System..::..StringOn entry: indicates how a starting basis (and certain other items) will be obtained.
Constraint: , or .
- Requests that an internal Crash procedure be used to choose an initial basis, unless a Basis file is provided via optional parameters Old Basis File, Insert File or Load File.
- Is the same as but is more meaningful when a Basis file is given.
- Means that a basis is already defined in hs and a start point is already defined in x (probably from an earlier call).
- Type: NagLibrary..::..E04..::..E04NQ_QPHXFor QP problems, you must supply a version of qphx to compute the matrix product for a given vector . If has rows and columns of zeros, it is most efficient to order so that the nonlinear variables appear first. For example, if and only enters the objective quadratically thenIn this case, ncolh should be the dimension of , and qphx should compute . For FP and LP problems, qphx will never be called by e04nq and hence qphx may be the dummy method E04NSH.
A delegate of type E04NQ_QPHX.
- Type: System..::..Int32On entry: , the number of general linear constraints (or slacks). This is the number of rows in the linear constraint matrix , including the free row (if any; see iobj). Note that must have at least one row. If your problem has no constraints, or only upper or lower bounds on the variables, then you must include a dummy row with sufficiently wide upper and lower bounds (see also acol, inda and loca).Constraint: .
- Type: System..::..Int32On entry: , the number of variables (excluding slacks). This is the number of columns in the linear constraint matrix .Constraint: .
- Type: System..::..Int32On entry: the number of nonzero elements in .Constraint: .
- Type: System..::..Int32On entry: the number of column (i.e., variable) and row names supplied in the array names.
Constraint: or .
- There are no names. Default names will be used in the printed output.
- All names must be supplied.
- Type: System..::..Int32On entry: the number of elements in the constant objective vector .If , the first lenc elements of belong to variables corresponding to the constant objective term .Constraint: .
- Type: System..::..Int32On entry: , the number of leading nonzero columns of the Hessian matrix . For FP and LP problems, ncolh must be set to zero.The first ncolh elements of belong to variables corresponding to the nonzero block of the QP Hessian.Constraint: .
- Type: System..::..Int32On entry: if , row iobj of is a free row containing the nonzero elements of the vector appearing in the linear objective term .If , there is no free row, and the linear objective vector should be supplied in array c.Constraint: .
- Type: System..::..DoubleOn entry: the constant , to be added to the objective for printing purposes. Typically .
- Type: System..::..StringOn entry: the name for the problem. It is used in the printed solution and in some methods that output Basis files. A blank name may be used.
- Type: array<System..::..Double>()An array of size [ne]On entry: the nonzero elements of , ordered by increasing column index. Note that all elements must be assigned a value in the calling program.
- Type: array<System..::..Int32>()An array of size [ne]On entry: must contain the row index of the nonzero element stored in , for . Thus a pair of values contains a matrix element and its corresponding row index.Note that the row indices for a column may be supplied in any order.Constraint: , for .
- Type: array<System..::..Int32>()An array of size On entry: must contain the index in acol and inda of the start of the th column, for . Thus for , the entries of column are held in and their corresponding row indices are in , where and . To specify the th column as empty, set . Note that the first and last elements of loca must be and . If your problem has no constraints, or just bounds on the variables, you may include a dummy ‘free’ row with a single (zero) element by setting , , , , and , for . This row is made ‘free’ by setting its bounds to be and , where is the value of the optional parameter Infinite Bound Size.Constraints:
- , for ;
- , for .
- Type: array<System..::..Double>()An array of size On entry: , the lower bounds for all the variables and general constraints, in the following order. The first n elements of bl must contain the bounds on the variables , and the next m elements the bounds for the general linear constraints (which, equivalently, are the bounds for the slacks, ) and the free row (if any). To fix the th variable, set , say, where . To specify a nonexistent lower bound (i.e., ), set . Here, is the value of the optional parameter Infinite Bound Size. To specify the th constraint as an equality, set , say, where . Note that the lower bound corresponding to the free row must be set to and stored in .Constraint: if ,(See also the description for bu.)
- Type: array<System..::..Double>()An array of size On entry: , the upper bounds for all the variables and general constraints, in the following order. The first n elements of bu must contain the bounds on the variables , and the next m elements the bounds for the general linear constraints (which, equivalently, are the bounds for the slacks, ) and the free row (if any). To specify a nonexistent upper bound (i.e., ), set . Note that the upper bound corresponding to the free row must be set to and stored in .Constraints:
- if , ;
- otherwise .
- Type: array<System..::..Double>()An array of size Note: the dimension of the array c must be at least if , and at least otherwise.
- Type: array<System..::..String>()An array of size [nname]On entry: the optional column and row names, respectively.If , names is not referenced and the printed output will use default names for the columns and rows.
- Type: array<System..::..Int32>()An array of size On entry: defines which variables are to be treated as being elastic in elastic mode. The allowed values of helast are:helast need not be assigned if optional parameter .
Status in elastic mode Variable is non-elastic and cannot be infeasible Variable can violate its lower bound Variable can violate its upper bound Variable can violate either its lower or upper boundConstraint: if , , for .
- Type: array<System..::..Int32>()An array of size On entry: if or , and a Basis file of some sort is to be input (see the description of the optional parameters Old Basis File, Insert File or Load File), then hs and x need not be set at all.If or and there is no Basis file, the first n elements of hs and x must specify the initial states and values, respectively, of the variables . (The slacks need not be initialized.) An internal Crash procedure is then used to select an initial basis matrix . The initial basis matrix will be triangular (neglecting certain small elements in each column). It is chosen from various rows and columns of . Possible values for are as follows:
State of during Crash procedure or Eligible for the basis Ignored Eligible for the basis (given preference over or ) or IgnoredIf nothing special is known about the problem, or there is no wish to provide special information, you may set and , for . All variables will then be eligible for the initial basis. Less trivially, to say that the th variable will probably be equal to one of its bounds, set and or and as appropriate.Following the Crash procedure, variables for which are made superbasic. Other variables not selected for the basis are then made nonbasic at the value if , or at the value or closest to .Constraints:
On exit: the final states of the variables and slacks . The significance of each possible value of is as follows:
- if or , , for ;
- if , , for .
State of variable Normal value of Nonbasic Nonbasic Superbasic Between and Basic Between andIf , basic and superbasic variables may be outside their bounds by as much as the value of the optional parameter Feasibility Tolerance. Note that unless the optional parameter is specified, the optional parameter Feasibility Tolerance applies to the variables of the scaled problem. In this case, the variables of the original problem may be as much as outside their bounds, but this is unlikely unless the problem is very badly scaled.Very occasionally some nonbasic variables may be outside their bounds by as much as the optional parameter Feasibility Tolerance, and there may be some nonbasic variables for which lies strictly between its bounds.If , some basic and superbasic variables may be outside their bounds by an arbitrary amount (bounded by sinf if ).
- Type: array<System..::..Double>()An array of size On entry: the initial values of the variables , and, if , the slacks , i.e., . (See the description for parameter hs.)On exit: the final values of the variables and slacks .
- Type: array<System..::..Double>()An array of size [m]On exit: contains the dual variables (a set of Lagrange multipliers (shadow prices) for the general constraints).
- Type: array<System..::..Double>()An array of size On exit: contains the reduced costs, . The vector is the gradient of the objective if x is feasible, otherwise it is the gradient of the Phase 1 objective. In the former case, , for , hence .
- Type: System..::..Int32%On entry: , the number of superbasics. For QP problems, ns need not be specified if , but must retain its value from a previous call when . For FP and LP problems, ns need not be initialized.On exit: the final number of superbasics. This will be zero for FP and LP problems.
- Type: System..::..Int32%On exit: the number of infeasibilities.
- Type: System..::..Double%On exit: the sum of the scaled infeasibilities. This will be zero if , and is most meaningful when .
- Type: System..::..Double%
- Type: NagLibrary..::..E04..::..e04nqOptionsAn Object of type E04.e04nqOptions. Used to configure optional parameters to this method.
e04nq is designed to solve large-scale linear or quadratic programming problems of the form:
where is an -vector of variables, and are constant lower and upper bounds, is an by sparse matrix and is a linear or quadratic objective function that may be specified in a variety of ways, depending upon the particular problem being solved. The optional parameter Maximize may be used to specify a problem in which is maximized instead of minimized.
Upper and lower bounds are specified for all variables and constraints. This form allows full generality in specifying various types of constraint. In particular, the th constraint may be defined as an equality by setting . If certain bounds are not present, the associated elements of or may be set to special values that are treated as or .
The possible forms for the function are summarised in Table 1. The most general form for is
where is a constant, is a constant -vector and is a constant symmetric by matrix with elements . In this form, is a quadratic function of and (1) is known as a quadratic program (QP). e04nq is suitable for all convex quadratic programs. The defining feature of a convex QP is that the matrix must be positive semidefinite, i.e., it must satisfy for all . If not, is nonconvex and e04nq will terminate with the error indicator . If is nonconvex it may be more appropriate to call e04vh instead.
|Problem type||Objective function||Hessian matrix|
|QP||Symmetric positive semidefinite|
If , then and the problem is known as a linear program (LP). In this case, rather than defining an with zero elements, you can define to have no columns by setting (see [Parameters]).
If , , and , there is no objective function and the problem is a feasible point problem (FP), which is equivalent to finding a point that satisfies the constraints on . In the situation where no feasible point exists, several options are available for finding a point that minimizes the constraint violations (see the description of the optional parameter Elastic Mode).
e04nq is suitable for large LPs and QPs in which the matrix is sparse, i.e., when the number of zero elements is sufficiently large that it is worthwhile using algorithms which avoid computations and storage involving zero elements. The matrix is input to e04nq by means of the three array parameters acol, inda and loca. This allows you to specify the pattern of nonzero elements in .
e04nq exploits structure in by requiring to be defined implicitly in a method that computes the product for any given vector . In many cases, the product can be computed very efficiently for any given , e.g., may be a sparse matrix, or a sum of matrices of rank-one.
For problems in which can be treated as a dense matrix, it is usually more efficient to use e04mf, e04nc or e04nf.
There is considerable flexibility allowed in the definition of in Table 1. The vector defining the linear term can be input in three ways: as a sparse row of ; as an explicit dense vector ; or as both a sparse row and an explicit vector (in which case, will be the sum of two linear terms). When stored in , is the iobjth row of , which is known as the objective row. The objective row must always be a free row of in the sense that its lower and upper bounds must be and . Storing as part of is recommended if is a sparse vector. Storing as an explicit vector is recommended for a sequence of problems, each with a different objective (see parameters c and lenc).
The upper and lower bounds on the elements of are said to define the general constraints of the problem. Internally, e04nq converts the general constraints to equalities by introducing a set of slack variables , where . For example, the linear constraint is replaced by , together with the bounded slack . The problem defined by (1) can therefore be re-written in the following equivalent form:
Since the slack variables are subject to the same upper and lower bounds as the elements of , the bounds on and can simply be thought of as bounds on the combined vector . (In order to indicate their special role in QP problems, the original variables are sometimes known as ‘column variables’, and the slack variables are known as ‘row variables’.)
Each LP or QP problem is solved using a two-phase iterative procedure (in which the general constraints are satisfied throughout): a feasibility phase (Phase 1), in which the sum of infeasibilities with respect to the bounds on and is minimized to find a feasible point that satisfies all constraints within a specified feasibility tolerance; and an optimality phase (Phase 2), in which is minimized (or maximized) by constructing a sequence of iterates that lies within the feasible region.
Phase 1 involves solving a linear program of the form
which is equivalent to minimizing the sum of the constraint violations. If the constraints are feasible (i.e., at least one feasible point exists), eventually a point will be found at which both and are zero. Then the associated value of satisfies the original constraints and is used as the starting point for the Phase 2 iterations for minimizing .
If the constraints are infeasible (i.e., or at the end of Phase 1), no solution exists for (1) and you have the option of either terminating or continuing in so-called elastic mode (see the discussion of the optional parameter Elastic Mode). In elastic mode, a ‘relaxed’ or ‘perturbed’ problem is solved in which is minimized while allowing some of the bounds to become ‘elastic’, i.e., to change from their specified values. Variables subject to elastic bounds are known as elastic variables. An elastic variable is free to violate one or both of its original upper or lower bounds. You are able to assign which bounds will become elastic if elastic mode is ever started (see the parameter helast in [Parameters]).
To make the relaxed problem meaningful, e04nq minimizes while (in some sense) finding the ‘smallest’ violation of the elastic variables. In the situation where all the variables are elastic, the relaxed problem has the form
where is a non-negative parameter known as the elastic weight (see the description of the optional parameter Elastic Weight), and is called the composite objective. In the more general situation where only a subset of the bounds are elastic, the 's and 's for the non-elastic bounds are fixed at zero.
|Phase 2 ()|
The elastic weight can be chosen to make the composite objective behave like the original objective , the sum of infeasibilities, or anything in-between. If , e04nq will attempt to minimize subject to the (true) upper and lower bounds on the non-elastic variables (and declare the problem infeasible if the non-elastic variables cannot be made feasible).
At the other extreme, choosing sufficiently large will have the effect of minimizing the sum of the violations of the elastic variables subject to the original constraints on the non-elastic variables. Choosing a large value of the elastic weight is useful for defining a ‘least-infeasible’ point for an infeasible problem.
In Phase 1 and elastic mode, all calculations involving and are done implicitly in the sense that an elastic variable is allowed to violate its lower bound (say) and an explicit value of can be recovered as .
A constraint is said to be active or binding at if the associated element of either or is equal to one of its upper or lower bounds. Since an active constraint in has its associated slack variable at a bound, the status of both simple and general upper and lower bounds can be conveniently described in terms of the status of the variables . A variable is said to be nonbasic if it is temporarily fixed at its upper or lower bound. It follows that regarding a general constraint as being active is equivalent to thinking of its associated slack as being nonbasic.
At each iteration of an active-set method, the constraints are (conceptually) partitioned into the form
where consists of the nonbasic elements of and the basis matrix is square and nonsingular. The elements of and are called the basic and superbasic variables respectively; with they are a permutation of the elements of and . At a QP solution, the basic and superbasic variables will lie somewhere between their upper or lower bounds, while the nonbasic variables will be equal to one of their bounds. At each iteration, is regarded as a set of independent variables that are free to move in any desired direction, namely one that will improve the value of the objective function (or sum of infeasibilities). The basic variables are then adjusted in order to ensure that continues to satisfy . The number of superbasic variables ( say) therefore indicates the number of degrees of freedom remaining after the constraints have been satisfied. In broad terms, is a measure of how nonlinear the problem is. In particular, will always be zero for FP and LP problems.
If it appears that no improvement can be made with the current definition of , and , a nonbasic variable is selected to be added to , and the process is repeated with the value of increased by one. At all stages, if a basic or superbasic variable encounters one of its bounds, the variable is made nonbasic and the value of is decreased by one.
Associated with each of the equality constraints is a dual variable . Similarly, each variable in has an associated reduced gradient (also known as a reduced cost). The reduced gradients for the variables are the quantities , where is the gradient of the QP objective function, and the reduced gradients for the slack variables are the dual variables . The QP subproblem is optimal if for all nonbasic variables at their lower bounds, for all nonbasic variables at their upper bounds and for all superbasic variables. In practice, an approximate QP solution is found by slightly relaxing these conditions on (see the description of the optional parameter Optimality Tolerance).
The process of computing and comparing reduced gradients is known as pricing (a term first introduced in the context of the simplex method for linear programming). To ‘price’ a nonbasic variable means that the reduced gradient associated with the relevant active upper or lower bound on is computed via the formula , where is the th column of . (The variable selected by such a process and the corresponding value of (i.e., its reduced gradient) are the quantities +SBS and dj in the monitoring file output; see [Description of the Printed Output].) If has significantly more columns than rows (i.e., ), pricing can be computationally expensive. In this case, a strategy known as partial pricing can be used to compute and compare only a subset of the s.
e04nq is based on SQOPT, which is part of the SNOPT package described in Gill et al. (2005a). It uses stable numerical methods throughout and includes a reliable basis package (for maintaining sparse factors of the basis matrix ), a practical anti-degeneracy procedure, efficient handling of linear constraints and bounds on the variables (by an active-set strategy), as well as automatic scaling of the constraints. Further details can be found in [Algorithmic Details].
Fourer R (1982) Solving staircase linear programs by the simplex method Math. Programming 23 274–313
Gill P E and Murray W (1978) Numerically stable methods for quadratic programming Math. Programming 14 349–372
Gill P E, Murray W and Saunders M A (1995) User's guide for QPOPT 1.0: a Fortran package for quadratic programming Report SOL 95-4 Department of Operations Research, Stanford University
Gill P E, Murray W and Saunders M A (2005a) Users' guide for SQOPT 7: a Fortran package for large-scale linear and quadratic programming Report NA 05-1 Department of Mathematics, University of California, San Diego http://www.ccom.ucsd.edu/~peg/papers/sqdoc7.pdf
Gill P E, Murray W and Saunders M A (2005b) Users' guide for SNOPT 7.1: a Fortran package for large-scale linear nonlinear programming Report NA 05-2 Department of Mathematics, University of California, San Diego http://www.ccom.ucsd.edu/~peg/papers/sndoc7.pdf
Gill P E, Murray W, Saunders M A and Wright M H (1987) Maintaining LU factors of a general sparse matrix Linear Algebra and its Applics. 88/89 239–270
Gill P E, Murray W, Saunders M A and Wright M H (1989) A practical anti-cycling procedure for linearly constrained optimization Math. Programming 45 437–474
Gill P E, Murray W, Saunders M A and Wright M H (1991) Inertia-controlling methods for general quadratic programming SIAM Rev. 33 1–36
Hall J A J and McKinnon K I M (1996) The simplest examples where the simplex method cycles and conditions where EXPAND fails to prevent cycling Report MS 96–100 Department of Mathematics and Statistics, University of Edinburgh
Note: e04nq may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the method:
- An input parameter is invalid.
- The requested accuracy could not be achieved.
- Weak QP solution found. The final is not unique.This exit will occur when the following are true:
(i) the problem is feasible; (ii) the reduced gradient is negligible; (iii) the Lagrange multipliers are optimal; and (iv) the reduced Hessian is singular or there are some very small multipliers.This exit cannot occur if is positive definite (i.e., is strictly convex).
- The problem is infeasible. The general constraints cannot all be satisfied simultaneously to within the value of the optional parameter Feasibility Tolerance.Feasibility is measured with respect to the upper and lower bounds on the variables and slacks. The message tells us that among all the points satisfying the general constraints , there is apparently no point that satisfies the bounds on and . Violations as small as the Feasibility Tolerance are ignored, but at least one component of or violates a bound by more than the tolerance.Note: although the objective function is the sum of infeasibilities (when ), this sum will not necessarily have been minimized when .
- The problem is unbounded (or badly scaled). For a minimization problem, the objective function is not bounded below in the feasible region.For linear problems, unboundedness is detected by the simplex method when a nonbasic variable can be increased or decreased by an arbitrary amount without causing a basic variable to violate a bound. Consider adding an upper or lower bound to the variable. Also, examine the constraints that have nonzeros in the associated column, to see if they have been formulated as intended.Very rarely, the scaling of the problem could be so poor that numerical error will give an erroneous indication of unboundedness. Consider using the optional parameter Scale Option.
- Too many iterations. The value of the optional parameter Iterations Limit is too small.The Iterations limit was exceeded before the required solution could be found. Check the iteration log to be sure that progress was being made. If so, restart the run using a Basis file that was saved at the end of the run.
- The value of the optional parameter Superbasics Limit is too small. The current set of basic and superbasic variables have been optimized as much as possible and a pricing operation is necessary to continue, but there are already Superbasics Limit superbasics (and no room for any more).In general, raise the Superbasics Limit by a reasonable amount, bearing in mind the storage needed for reduced Hessian (see [Definition of the Working Set and Search Direction]). (The Reduced Hessian Dimension will also increase to unless specified otherwise, and the associated storage will be about words.) In some cases you may have to set to conserve storage, but beware that the rate of convergence will probably fall off severely.
- The basis is singular after several attempts to factorize it (adding slacks where necessary). Either the problem is badly scaled or the value of the optional parameter LU Factor Tolerance is too large.
- Numerical error in trying to satisfy the general constraints. The basis is very ill-conditioned.An factorization of the basis has just been obtained and used to recompute the basic variables , given the present values of the superbasic and nonbasic variables. However, a row check has revealed that the resulting solution does not satisfy the current constraints sufficiently well.This probably means that the current basis is very ill-conditioned. Request the Scale Option if there are any linear constraints and variables.For certain highly structured basis matrices (notably those with band structure), a systematic growth may occur in the factor . Consult the description of Umax, Umin and Growth in [Description of Monitoring Information], and set the optional parameter LU Factor Tolerance to (or possibly even smaller, but not less than ).
- An indefinite matrix was detected during the computation of the reduced Hessian factor (see [Definition of the Working Set and Search Direction]). This may be caused by being indefinite. Check also that qphx has been coded correctly and that all relevant elements of have been assigned their correct values. If qphx is coded correctly and is positive semidefinite, the failure may be caused by ill conditioning. Try reducing the values of the optional parameters LU Factor Tolerance and LU Update Tolerance. If there are very large values in , check the scaling of the variables and constraints.
- Internal memory allocation failed when attempting to obtain the required workspace. Please contact NAG.
- Internal memory allocation was insufficient. Please contact NAG.
- An unexpected error has occurred. Set the optional parameter Print File and examine the output carefully for further information.
e04nq implements a numerically stable active-set strategy and returns solutions that are as accurate as the condition of the problem warrants on the machine.
This section contains a description of the printed output.
If , one line of information is output to the Print File every th iteration, where is the specified Print Frequency. A heading is printed before the first such line following a basis factorization. The heading contains the items described below. In this description, a pricing operation is defined to be the process by which one or more nonbasic variables are selected to become superbasic (in addition to those already in the superbasic set). The variable selected will be denoted by jq. If the problem is purely linear, variable jq will usually become basic immediately (unless it should happen to reach its opposite bound and return to the nonbasic set).
If optional parameter Partial Price is in effect, variable jq is selected from or , the ppth segments of the constraint matrix .
|Itn||is the iteration count.|
|pp||is the partial-price indicator. The variable selected by the last pricing operation came from the ppth partition of and . Note that pp is reset to zero whenever the basis is refactorized.|
|dj||is the value of the reduced gradient (or reduced cost) for the variable selected by the pricing operation at the start of the current iteration.
Algebraically, dj is , for , where is the gradient of the current objective function, is the vector of dual variables, and is the th column of the constraint matrix .
Note that dj is the norm of the reduced-gradient vector at the start of the iteration, just after the pricing operation.
|+SBS||is the variable jq selected by the pricing operation to be added to the superbasic set.|
|-SBS||is the variable chosen to leave the superbasic set. It has become basic if the entry under -B is nonzero, otherwise it becomes nonbasic.|
|-BS||is the variable removed from the basis to become nonbasic.|
|Step||is the value of the step length taken along the current search direction . The variables have just been changed to . If a variable is made superbasic during the current iteration (i.e., +SBS is positive), Step will be the step to the nearest bound. During the optimality phase, the step can be greater than unity only if the reduced Hessian is not positive definite.|
|Pivot||is the th element of a vector satisfying whenever (the th column of the constraint matrix replaces the th column of the basis matrix . Wherever possible, Step is chosen so as to avoid extremely small values of Pivot (since they may cause the basis to be nearly singular). In extreme cases, it may be necessary to increase the value of the optional parameter Pivot Tolerance to exclude very small elements of from consideration during the computation of Step.|
|nInf||is the number of violated constraints (infeasibilities) before the present iteration. This number will not increase unless iterations are in elastic mode.|
|sInf||is the sum of infeasibilities before the present iteration. It will usually decrease at each nonzero step, but if nInf decreases by or more, sInf may occasionally increase. However, in elastic mode it will decrease monotonically.|
|Objective||is the value of the current objective function after the present iteration. Note, if Elastic Mode is , the heading is Composite Obj.|
|L+U||L is the number of nonzeros in the basis factor . Immediately after a basis factorization , L contains lenL (see [Description of Monitoring Information]). Further nonzeros are added to L when various columns of are later replaced. (Thus, L increases monotonically.) U is the number of nonzeros in the basis factor . Immediately after a basis factorization , U contains lenU (see [Description of Monitoring Information]). As columns of are replaced, the matrix is maintained explicitly (in sparse form). The value of U may fluctuate up or down; in general, it will tend to increase.|
|ncp||is the number of compressions required to recover workspace in the data structure for . This includes the number of compressions needed during the previous basis factorization. Normally, ncp should increase very slowly.|
The following will be output if the problem is QP or if the superbasic set is non-empty.
|rgNorm||is the largest reduced-gradient among the superbasic variables after the current iteration. During the optimality phase, this will be approximately zero after a unit step.|
|nS||is the current number of superbasic variables.|
|condHz||is a lower bound on the condition number of the reduced Hessian (see [Definition of the Working Set and Search Direction]). The larger this number, the more difficult the problem. Attention should be given to the scaling of the variables and the constraints to guard against high values of condHz.|
This example minimizes the quadratic function , where
subject to the bounds
and to the linear constraints
The initial point, which is infeasible, is
The optimal solution (to five figures) is
One bound constraint and four linear constraints are active at the solution. Note that the Hessian matrix is positive semidefinite.
This section contains a detailed description of the method used by e04nq.
e04nq is based on an inertia-controlling method that maintains a Cholesky factorization of the reduced Hessian (see below). The method is similar to that of Gill and Murray (1978), and is described in detail by Gill et al. (1991). Here we briefly summarise the main features of the method. Where possible, explicit reference is made to the names of variables that are parameters of the method or appear in the printed output.
The method used has two distinct phases: finding an initial feasible point by minimizing the sum of infeasibilities (the feasibility phase), and minimizing the quadratic objective function within the feasible region (the optimality phase). The computations in both phases are performed by the same methods. The two-phase nature of the algorithm is reflected by changing the function being minimized from the sum of infeasibilities (the printed quantity sInf; see [Description of the Printed Output]) to the quadratic objective function (the printed quantity Objective; see [Description of the Printed Output]).
In general, an iterative process is required to solve a quadratic program. Given an iterate in both the original variables and the slack variables , a new iterate is defined by
where the step length is a non-negative scalar (the printed quantity Step; see [Description of Monitoring Information]), and is called the search direction. (For simplicity, we shall consider a typical iteration and avoid reference to the index of the iteration.) Once an iterate is feasible (i.e., satisfies the constraints), all subsequent iterates remain feasible.
At each iterate , a working set of constraints is defined to be a linearly independent subset of the constraints that are satisfied ‘exactly’ (to within the value of the optional parameter Feasibility Tolerance). The working set is the current prediction of the constraints that hold with equality at a solution of the LP or QP problem. Let denote the number of constraints in the working set (including bounds), and let denote the associated by working set matrix consisting of the gradients of the working set constraints.
The search direction is defined so that constraints in the working set remain unaltered for any value of the step length. It follows that must satisfy the identity
This characterisation allows to be computed using any by full-rank matrix that spans the null space of . (Thus, and .) The null space matrix is defined from a sparse factorization of part of (see (7) and (8)). The direction will satisfy (4) if
where is any -vector.
The working set contains the constraints and a subset of the upper and lower bounds on the variables . Since the gradient of a bound constraint or is a vector of all zeros except for in position , it follows that the working set matrix contains the rows of and the unit rows associated with the upper and lower bounds in the working set.
The working set matrix can be represented in terms of a certain column partition of the matrix by (conceptually) partitioning the constraints so that
where is a square nonsingular basis and , and are the basic, superbasic and nonbasic variables respectively. The nonbasic variables are equal to their upper or lower bounds at , and the superbasic variables are independent variables that are chosen to improve the value of the current objective function. The number of superbasic variables is (the printed quantity nS; see [Description of the Printed Output]). Given values of and , the basic variables are adjusted so that satisfies (6).
If is a permutation matrix such that , then satisfies
where is the identity matrix with the same number of columns as .
The null space matrix is defined from a sparse factorization of part of . In particular, is maintained in ‘reduced gradient’ form, using the LUSOL package (see Gill et al. (1991)) to maintain sparse factors of the basis matrix as the partition changes. Given the permutation , the null space basis is given by
This matrix is used only as an operator, i.e., it is never computed explicitly. Products of the form and are obtained by solving with or . This choice of implies that , the number of ‘degrees of freedom’ at , is the same as , the number of superbasic variables.
Let and denote the reduced gradient and reduced Hessian of the objective function:
where is the objective gradient at . Roughly speaking, and describe the first and second derivatives of an -dimensional unconstrained problem for the calculation of . (The condition estimator of is the quantity condHz in the monitoring file output; see [Description of the Printed Output].)
At each iteration, an upper triangular factor is available such that . Normally, is computed from at the start of the optimality phase and then updated as the QP working set changes. For efficiency, the dimension of should not be excessive (say, ). This is guaranteed if the number of nonlinear variables is ‘moderate’.
If the QP problem contains linear variables, is positive semidefinite and may be singular with at least one zero diagonal element. However, an inertia-controlling strategy is used to ensure that only the last diagonal element of can be zero. (See Gill et al. (1991) for a discussion of a similar strategy for indefinite quadratic programming.)
If the initial is singular, enough variables are fixed at their current value to give a nonsingular . This is equivalent to including temporary bound constraints in the working set. Thereafter, can become singular only when a constraint is deleted from the working set (in which case no further constraints are deleted until becomes nonsingular).
If the reduced gradient is zero, is a constrained stationary point on the working set. During the feasibility phase, the reduced gradient will usually be zero only at a vertex (although it may be zero elsewhere in the presence of constraint dependencies). During the optimality phase, a zero reduced gradient implies that minimizes the quadratic objective function when the constraints in the working set are treated as equalities. At a constrained stationary point, Lagrange multipliers are defined from the equations
A Lagrange multiplier, , corresponding to an inequality constraint in the working set is said to be optimal if when the associated constraint is at its upper bound, or if when the associated constraint is at its lower bound, where depends on the value of the optional parameter Optimality Tolerance. If a multiplier is nonoptimal, the objective function (either the true objective or the sum of infeasibilities) can be reduced by continuing the minimization with the corresponding constraint excluded from the working set. (This step is sometimes referred to as ‘deleting’ a constraint from the working set.) If optimal multipliers occur during the feasibility phase but the sum of infeasibilities is nonzero, there is no feasible point and the method terminates immediately with .
The special form (7) of the working set allows the multiplier vector , the solution of (10), to be written in terms of the vector
where satisfies the equations , and denotes the basic elements of . The elements of are the Lagrange multipliers associated with the equality constraints . The vector of nonbasic elements of consists of the Lagrange multipliers associated with the upper and lower bound constraints in the working set. The vector of superbasic elements of is the reduced gradient in (9). The vector of basic elements of is zero, by construction. (The Euclidean norm of and the final values of , and are the quantities rgNorm, Reduced Gradnt, Obj Gradient and Dual Activity in the monitoring file output; see [Description of Monitoring Information].)
If the reduced gradient is not zero, Lagrange multipliers need not be computed and the search direction is given by (see (8) and (12)). The step length is chosen to maintain feasibility with respect to the satisfied constraints.
There are two possible choices for , depending on whether or not is singular. If is nonsingular, is nonsingular and in (5) is computed from the equations
where is the reduced gradient at . In this case, is the minimizer of the objective function subject to the working set constraints being treated as equalities. If is feasible, is defined to be unity. In this case, the reduced gradient at will be zero, and Lagrange multipliers are computed at the next iteration. Otherwise, is set to , the step to the ‘nearest’ constraint along . This constraint is then added to the working set at the next iteration.
If is singular, then must also be singular, and an inertia-controlling strategy is used to ensure that only the last diagonal element of is zero. (See Gill et al. (1991) for a discussion of a similar strategy for indefinite quadratic programming.) In this case, satisfies
which allows the objective function to be reduced by any step of the form , where . The vector is a direction of unbounded descent for the QP problem in the sense that the QP objective is linear and decreases without bound along . If no finite step of the form (where ) reaches a constraint not in the working set, the QP problem is unbounded and the method terminates immediately with . Otherwise, is defined as the maximum feasible step along and a constraint active at is added to the working set for the next iteration.
e04nq makes explicit allowance for infeasible constraints. Infeasible linear constraints are detected first by solving a problem of the form
where . This is equivalent to minimizing the sum of the general linear constraint violations subject to the simple bounds. (In the linear programming literature, the approach is often called elastic programming.)
If the basis matrix is not chosen carefully, the condition of the null space matrix in (8) could be arbitrarily high. To guard against this, the method implements a ‘basis repair’ feature in which the LUSOL package (see Gill et al. (1991)) is used to compute the rectangular factorization
returning just the permutation that makes unit lower triangular. The pivot tolerance is set to require , and the permutation is used to define in (7). It can be shown that is likely to be little more than unity. Hence, should be well-conditioned regardless of the condition of . This feature is applied at the beginning of the optimality phase if a potential ordering is known.
The EXPAND procedure (see Gill et al. (1989)) is used to reduce the possibility of cycling at a point where the active constraints are nearly linearly dependent. Although there is no absolute guarantee that cycling will not occur, the probability of cycling is extremely small (see Hall and McKinnon (1996)). The main feature of EXPAND is that the feasibility tolerance is increased at the start of every iteration. This allows a positive step to be taken at every iteration, perhaps at the expense of violating the bounds on by a small amount.
Suppose that the value of the optional parameter Feasibility Tolerance is . Over a period of iterations (where is the value of the optional parameter Expand Frequency), the feasibility tolerance actually used by the method (i.e., the working feasibility tolerance) increases from to (in steps of ).
At certain stages the following ‘resetting procedure’ is used to remove small constraint infeasibilities. First, all nonbasic variables are moved exactly onto their bounds. A count is kept of the number of nontrivial adjustments made. If the count is nonzero, the basic variables are recomputed. Finally, the working feasibility tolerance is reinitialized to .
If a problem requires more than iterations, the resetting procedure is invoked and a new cycle of iterations is started. (The decision to resume the feasibility phase or optimality phase is based on comparing any constraint infeasibilities with .)
The resetting procedure is also invoked when the method reaches an apparently optimal, infeasible or unbounded solution, unless this situation has already occurred twice. If any nontrivial adjustments are made, iterations are continued.
The EXPAND procedure not only allows a positive step to be taken at every iteration, but also provides a potential choice of constraints to be added to the working set. All constraints at a distance (where ) along from the current point are then viewed as acceptable candidates for inclusion in the working set. The constraint whose normal makes the largest angle with the search direction is added to the working set. This strategy helps keep the basis matrix well-conditioned.
This section describes the intermediate printout and final printout which constitutes the monitoring information produced by e04nq. (See also the description of the optional parameters Print File and Print Level.) You can control the level of printed output.
When , and System Information Yes has been specified, the following lines of intermediate printout (less than characters) are produced on the unit number specified by optional parameter Print File whenever (see [Parameters]). They refer to the number of columns selected by the Crash procedure during each of several passes through , whilst searching for a triangular basis matrix.
|Slacks||is the number of slacks selected initially.|
|Free cols||is the number of free columns in the basis, including those whose bounds are rather far apart.|
|Preferred||is the number of ‘preferred’ columns in the basis (i.e., for some ). It will be a subset of the columns for which was specified.|
|Unit||is the number of unit columns in the basis.|
|Double||is the number of double columns in the basis.|
|Triangle||is the number of triangular columns in the basis.|
|Pad||is the number of slacks used to pad the basis (to make it a nonsingular triangle).|
When and , the first seven items of intermediate printout in the list below are produced on the unit number specified by optional parameter Print File whenever the matrix or is factorized. Gaussian elimination is used to compute an factorization of or , where is a lower triangular matrix and is an upper triangular matrix for some permutation matrices and . The factorization is stabilized in the manner described under the optional parameter LU Factor Tolerance. In addition, if System Information Yes has been specified, the entries from Elems onwards are also output.
|Factor||the number of factorizations since the start of the run.|
|Demand||a code giving the reason for the present factorization, as follows:
|Itn||is the current minor iteration number.|
|Nonlin||is the number of nonlinear variables in the current basis .|
|Linear||is the number of linear variables in .|
|Slacks||is the number of slack variables in .|
|B, BR, BS or BT factorize||is the type of factorization.
|m||is the number of rows in or .|
|n||is the number of columns in or . Preceded by ‘=’ or ‘>’ respectively.|
|Elems||is the number of nonzero elements in or .|
|Amax||is the largest nonzero in or .|
|Density||is the percentage nonzero density of or .|
|Merit/MerRP/MerCP||Merit is the average Markowitz merit count for the elements chosen to be the diagonals of . Each merit count is defined to be where and are the number of nonzeros in the column and row containing the element at the time it is selected to be the next diagonal. Merit is the average of n such quantities. It gives an indication of how much work was required to preserve sparsity during the factorization. If LU Complete Pivoting or LU Rook Pivoting has been selected, this heading is changed to MerCP, respectively MerRP.|
|lenL||is the number of nonzeros in .|
|L+U||is the number of nonzeros representing the basis factors and . Immediately after a basis factorization , this is lenL+lenU, the number of subdiagonal elements in the columns of a lower triangular matrix and the number of diagonal and superdiagonal elements in the rows of an upper-triangular matrix. Further nonzeros are added to L when various columns of are later replaced. As columns of are replaced, the matrix is maintained explicitly (in sparse form). The value of L will steadily increase, whereas the value of U may fluctuate up or down. Thus the value of L+U may fluctuate up or down (in general, it will tend to increase).|
|Cmpressns||is the number of times the data structure holding the partially factored matrix needed to be compressed to recover unused storage. Ideally this number should be zero. If it is more than or , the amount of workspace available to e04nq should be increased for efficiency.|
|Incres||is the percentage increase in the number of nonzeros in and relative to the number of nonzeros in or .|
|Utri||is the number of triangular rows of or at the top of .|
|lenU||the number of nonzeros in , including its diagonals.|
|Ltol||is the largest subdiagonal element allowed in . This is the specified LU Factor Tolerance or a smaller value that is currently being used for greater stability.|
|Umax||the maximum nonzero element in .|
|Ugrwth||is the ratio , which ideally should not be substantially larger than or . If it is orders of magnitude larger, it may be advisable to reduce the LU Factor Tolerance to , , or , say (but bigger than ).
As long as Lmax is not large (say or less), gives an estimate of the condition number . If this is extremely large, the basis is nearly singular. Slacks are used to replace suspect columns of and the modified basis is refactored.
|Ltri||is the number of triangular columns of or at the left of .|
|dense1||is the number of columns remaining when the density of the basis matrix being factorized reached .|
|Lmax||is the actual maximum subdiagonal element in (bounded by Ltol).|
|Akmax||is the largest nonzero generated at any stage of the factorization. (Values much larger than Amax indicate instability.) Akmax is not printed if LU Partial Pivoting is selected.|
|Agrwth||is the ratio . Values much larger than (say) indicate instability. Agrwth is not printed if LU Partial Pivoting is selected.|
|bump||is the size of the block to be factorized nontrivially after the triangular rows and columns of or have been removed.|
|dense2||is the number of columns remaining when the density of the basis matrix being factorized reached . (The Markowitz pivot strategy searches fewer columns at that stage.)|
|DUmax||is the largest diagonal of .|
|DUmin||is the smallest diagonal of .|
|condU||the ratio , which estimates the condition number of (and of if Ltol is less than , say).|
When and , the following lines of intermediate printout (less than characters) are produced on the unit number specified by optional parameter Print File. They refer to the elements of the names array (see [Parameters]).
|Name||gives the name for the problem (blank if problem unnamed).|
|Infeasibilities||gives the number of infeasibilities. Printed only if the final point is infeasible.|
|Objective Value||gives the objective value at the final point (or the value of the sum of infeasibilities). Printed only if the final point is feasible.|
|Status||gives the exit status for the problem (i.e., Optimal soln, Weak soln, Unbounded, Infeasible, Excess itns, Error condn or Feasble soln) followed by details of the direction of the optimization (i.e., (Min) or (Max)).|
|Iteration||gives the iteration number when the file was created.|
|Superbasics||gives the number of superbasic variables.|
|Objective||gives the name of the free row for the problem (blank if objective unnamed).|
|RHS||gives the name of the constraint right-hand side for the problem (blank if objective unnamed).|
|Ranges||gives the name of the ranges for the problem (blank if objective unnamed).|
|Bounds||gives the name of the bounds for the problem (blank if objective unnamed).|
At the end of a run, the final solution will be output to the Print file. Some header information appears first to identify the problem and the final state of the optimization procedure. A ROWS section and a COLUMNS section then follow, giving one line of information for each row and column.
General constraints take the form . The th constraint is therefore of the form
where is the th row of .
Internally, the constraints take the form , where is the set of slack variables (which happen to satisfy the bounds ). For the th constraint, the slack variable is directly available, and it is sometimes convenient to refer to its state. It should satisfy . A fullstop (.) is printed for any numerical value that is exactly zero.
|Number||is the value of . (This is used internally to refer to in the intermediate output.)|
|Row||gives the name of .|
|State||the state of (the state of relative to the bounds and ). The various states possible are as follows:
A key is sometimes printed before State. Note that unless the optional parameter is specified, the tests for assigning a key are applied to the variables of the scaled problem.
|Activity||is the value of at the final iterate.|
|Slack Activity||is the value by which the row differs from its nearest bound. (For the free row (if any), it is set to Activity.)|
|Lower Limit||is , the lower bound specified for the variable . None indicates that .|
|Upper Limit||is , the upper bound specified for the variable . None indicates that .|
|Dual Activity||is the value of the dual variable (the Lagrange multiplier for ; see [Main Iteration]). For FP problems, is set to zero.|
|i||gives the index of the th row.|
Let the th component of be the variable and assume that it satisfies the bounds . A fullstop (.) is printed for any numerical value that is exactly zero.
|Number||is the column number . (This is used internally to refer to in the intermediate output.)|
|Column||gives the name of .|
|State||the state of relative to the bounds and . The various states possible are as follows:
A key is sometimes printed before State. Note that unless the optional parameter is specified, the tests for assigning a key are applied to the variables of the scaled problem.
|Activity||is the value of at the final iterate.|
|Obj Gradient||is the value of at the final iterate. For FP problems, is set to zero.|
|Lower Limit||is the lower bound specified for the variable. None indicates that .|
|Upper Limit||is the upper bound specified for the variable. None indicates that .|
|Reduced Gradnt||is the value of at the final iterate (see [Main Iteration]). For FP problems, is set to zero.|
|m + j||is the value of .|
Note: if two problems are the same except that one minimizes and the other maximizes , their solutions will be the same but the signs of the dual variables and the reduced gradients will be reversed.
If a positive Solution File is specified, the information contained in a printed solution may also be output to the relevant file (which may be the Print file if so desired). Infinite Upper and Lower limits appear as rather than None. Other real values are output with format 1pe16.6. The maximum line length is characters, including what would be the carriage-control character if the file were printed.
A Solution file is intended to be read from disk by a self-contained program that extracts and saves certain values as required for possible further computation. Typically the first lines would be ignored. Each subsequent line may be read using adapted to suit the occasion. The end of the ROWS section is marked by a line that starts with a and is otherwise blank. If this and the next lines are skipped, the COLUMNS section (see [The COLUMNS Section]) can then be read under the same format. (There should be no need to use any BACKSPACE statements.)
If Summary File is specified with , certain brief information will be output to unit . When e04nq is run interactively, unit will usually be the terminal. For batch jobs a disk file should be used, to retain a concise log of each run if desired. (A Summary File is more easily perused than the associated Print file).
A Summary file (like the Print file) is not rewound after a problem has been processed. The maximum line length is characters, including a carriage-control character in column 1.
The following information is included:
|1.||The optional parameters supplied via the option setting methods, if any;|
|2.||The Basis file loaded, if any;|
|3.||The status of the solution after each basis factorization (whether feasible; the objective value; the number of function calls so far);|
|4.||The same information every th iteration, where is the specified Summary Frequency;|
|5.||Warnings and error messages;|
|6.||The exit condition and a summary of the final solution.|
The meaning of the printout for linear constraints is the same as that given above for variables, with ‘variable’ replaced by ‘constraint’, replaced by , replaced by , and are replaced by and respectively, and with the following change in the heading:
|Constrnt||gives the name of the linear constraint.|
Note that movement off a constraint (as opposed to a variable moving away from its bound) can be interpreted as allowing the entry in the Residual column to become positive.
Numerical values are output with a fixed number of digits; they are not guaranteed to be accurate to this precision.