Degeneracy graphs: theory and application an updated survey

A survey of various aspects of the theory and application of degeneracy graphs (DGs for short) is given. The notion and some basic properties of DGs are introduced, cycling of the simplex method is discussed, the neighborhood problem is tackled, and the application of the so-called optimum DGs to particular problems which are connected with optimal degenerate solutions of a linear programming problem is presented. The impact of weakly redundant constraints on various postoptimal analyses under degeneracy is briefly described.     

On degeneracy in linear programming and related problems

Methods related to Wolfe's recursive method for the resolution of degeneracy in linear programming are discussed, and a nonrecursive variant which works with probability one suggested. Numerical results for both nondegenerate problems and problems constructed to have degenerate optima are reported. These are obtained using a careful implementation of the projected gradient algorithm [11]. These are compared with results obtained using a steepest descent approach which can be implemented by means of a closely related projected gradient method, and which is not affected by degeneracy in principle. However, the problem of correctly identifying degenerate active sets occurs with both algorithms. The numerical results favour the more standard projected gradient procedure which resolves the degeneracy explicitly. Extension of both methods to general polyhedral convex function minimization problems is sketched.     

Degeneracy: Resolve or Avoid?

Two projected gradient algorithms for linear programming are discussed. The first uses a conventional enough steepest edge approach, and implements a version of Wolfe's method for resolving any problems of degeneracy. The second makes use of a steepest descent direction which is calculated by solving a linear least squares problem subject to bound constraints, and avoids problems of degeneracy altogether. They are compared using randomly generated problems which permit the specification of (known) degenerate minima. The results appear to favour the more conventional method as problem sizes get larger.     

Row-reduced column generation for degenerate master problems

Column generation for solving linear programs with a huge number of variables alternates between solving a master problem and a pricing subproblem to add variables to the master problem as needed. The method is known to often suffer from degeneracy in the master problem. Inspired by recent advances in coping with degeneracy in the primal simplex method, we propose a row-reduced column generation method that may take advantage of degenerate solutions. The idea is to reduce the number of constraints to the number of strictly positive basic variables in the current master problem solution. The advantage of this row-reduction is a smaller working basis, and thus a faster re-optimization of the master problem. This comes at the expense of a more involved pricing subproblem, itself eventually solved by column generation, that needs to generate weighted subsets of variables that are said compatible with the row-reduction, if possible. Such a subset of variables gives rise to a strict improvement in the objective function value if the weighted combination of the reduced costs is negative. We thus state, as a by-product, a necessary and sufficient optimality condition for linear programming.     

An approach to find redundant objective function(s) and redundant constraint(s) in multi-objective nonlinear stochastic fractional programming problems

Structural redundancies in mathematical programming models are nothing uncommon and nonlinear programming problems are no exception. Over the past few decades numerous papers have been written on redundancy. Redundancy in constraints and variables are usually studied in a class of mathematical programming problems. However, main emphasis has so far been given only to linear programming problems. In this paper, an algorithm that identifies redundant objective function(s) and redundant constraint(s) simultaneously in multi-objective nonlinear stochastic fractional programming problems is provided. A solution procedure is also illustrated with numerical examples. The proposed algorithm reduces the number of nonlinear fractional objective functions and constraints in cases where redundancy exists.     

New approach to optimality conditions for degenerate nonlinear programming problems

In the work, a new approach to constructing optimality conditions for degenerate smooth optimization problems with inequality constraints is proposed. The approach is based on the theory of p-regularity. A special case of degeneracy, when the first derivatives of some function-constraints are equal to zero up to some order, is considered. Optimality conditions for the general case of degeneracy with p = 2 are presented. Proposed constructions and optimality conditions are illustrated by an example. A general case of degeneracy is considered and optimality conditions for the case of p ≥ 2 are proposed.     

Redundant objective functions in linear vector maximum problems and their determination

Suppose that in a multicriteria linear programming problem among the given objective functions there are some which can be deleted without influencing the set E of all efficient solutions. Such objectives are said to be redundant. Introducing systems of objective functions which realize their individual optimum in a single vertex of the polyhedron generated by the restriction set, the notion of relative or absolute redundant objectives is defined. A theory which describes properties of absolute and relative redundant objectives is developed. A method for determining all the relative and absolute redundant objectives, based on this theory, is given. Illustrative examples demonstrate the procedure.     

A dual approach to primal degeneracy

The Revised Primal Simplex algorithm, in its simplest form, has no defence against degeneracy. Various forms of the perturbation method are usually effective, but most offer no guarantee of avoiding all degeneracy, and can lead to numerical difficulties. This paper presents a method that avoids cycling and circling by taking a dual approach.The degenerate subproblem consists of all the original variables, but only the degenerate transformed constraints. The current primal objective, which may be mixed, is used. This subproblem may be solved using the dual simplex algorithm, starting from the current dual infeasible solution, and with a zero dual objective. If the dual algorithm terminates optimally then the whole problem is optimal (subject to primal feasibility). Otherwise the final solution provides a non-basic direction which improves the value of the mixed primal objective and moves away from the degenerate vertex. A purification algorithm then renders the solution basic and further improves the mixed objective.     

On degeneracy and collapsing in the construction of the set of objective values in a multiple objective linear program

In this paper an algorithm is developed to generate all nondominated extreme points and edges of the set of objective values of a multiple objective linear program. The approach uses simplex tableaux but avoids generating unnecessary extreme points or bases of extreme points. The procedure is based on, and improves, an algorithm Dauer and Liu developed for this problem. Essential to this approach is the work of Gal and Kruse on the neighborhood problem of determining all extreme points of a convex polytope that are adjacent to a given (degenerate) extreme point of the set. The algorithm will incorporate Gal's degeneracy graph approach to the neighborhood problem with Dauer's objective space analysis of multiple objective linear programs.     

An Algorithm for Solving Optimization Problems with One Linear Objective Function and Finitely Many Constraints of Fuzzy Relation Inequalities

An optimization model with one linear objective function and fuzzy relation equation constraints was presented by Fang and Li (1999) as well as an efficient solution procedure was designed by them for solving such a problem. A more general case of the problem, an optimization model with one linear objective function and finitely many constraints of fuzzy relation inequalities, is investigated in this paper. A new approach for solving this problem is proposed based on a necessary condition of optimality given in the paper. Compared with the known methods, the proposed algorithm shrinks the searching region and hence obtains an optimal solution fast. For some special cases, the proposed algorithm reaches an optimal solution very fast since there is only one minimum solution in the shrunk searching region. At the end of the paper, two numerical examples are given to illustrate this difference between the proposed algorithm and the known ones.     

A GENERAL TECHNIQUE FOR DEALING WITH DEGENERACY IN REDUCED GRADIENT METHODS FOR LINEARLY CONSTRAINED NONLINEAR PROGRAMMING

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A Note on Redundancy and Linear Parametric Programming

A simple redundancy criterion for a system of linear inequalities is given. Considering a linear programming problem with linear inequality constraints the sensitivity analysis of the optimal solution of the right-hand side in combination with the redundancy criterion can help to identify redundant inequalities (constraints). The redundancy criterion is applied to a linear parametric problem which is enlarged by an auxiliary condition such as "stop the parametric procedure when a chosen constraint becomes redundant".     

A degenerate extreme point strategy for the classification of linear constraints as redundant or necessary

This paper presents a degenerate extreme point strategy for active set algorithms which classify linear constraints as either redundant or necessary. The strategy makes use of an efficient method for classifying constraints active at degenerate extreme points. Numerical results indicate that significant savings in the computational effort required to classify the constraints can be achieved.This research was supported by the Natural Sciences and Engineering Research Council of Canada under Grants A8807 and A4625 and by an Undergraduate Summer Research Award.     

Resolving degeneracy in quadratic programming

A technique for the resolution of degeneracy in an Active Set Method for Quadratic Programming is described. The approach generalises Fletcher's method [2] which applies to the LP case. The method is described in terms of an LCP tableau, which is seen to provide useful insights. It is shown that the degeneracy procedure only needs to operate when the degenerate constraints are linearly dependent on those in the active set. No significant overheads are incurred by the degeneracy procedure. It is readily implemented in a null space format, and no complications in the matrix algebra are introduced.The guarantees of termination provided by [2], extending in particular to the case where round-off error is present, are preserved in the QP case. It is argued that the technique gives stronger guarantees than are available with other popular methods such as Wolfe's method [11] or the method of Goldfarb and Idnani [7].Presented at the 14th International Symposium on Mathematical Programming, Amsterdam, August 5–9, 1991.     

Degeneracy in NLP and the development of results motivated by its presence

We study notions of nondegeneracy and several levels of increasing degeneracy from the perspective of the local behavior of a local solution of a nonlinear program when problem parameters are slightly perturbed. Ideal nondegeneracy at a local minimizer is taken to mean satisfaction of second order sufficient conditions, linear independence and strict complimentary slackness. Following a brief exploration of the relationship of these conditions with the classical definition of nondegeneracy in linear programming, we recall a number of optimality and regularity conditions used to attempt to resolve degeneracy and survey results of Fiacco, McCormick, Robinson, Kojima, Gauvin and Janin, Shapiro, Kyparisis and Liu. This overview may be viewed as a structured survey of sensitivity and stability results: the focus is on progressive levels of degeneracy. We note connections of nondegeneracy with the convergence of algorithms and observe the striking parallel between the effects of nondegeneracy and degeneracy on optimality conditions, stability analysis and algorithmic convergence behavior. Although our orientation here is primarily interpretive and noncritical, we conclude that more effort is needed to unify optimality, stability and convergence theory and more results are needed in all three areas for radically degenerate problems.Research supported by National Science Foundation Grant ECS 90-00560 and Grant N00014-89-J-1537 Office of Naval Research     

Optimization of dual response systems: A comprehensive procedure for degenerate and nondegenerate problems

Most dual response systems (DRSs) arising in response surface modeling can be approximated using a nonlinear (and typically nonconvex) mathematical program involving two quadratic functions. One of the quadratic functions is used as the objective function, the other for imposing a target constraint. This paper describes an effective heuristic for computing global (or near-global) optimal solutions for this type of problem. The first part of the paper addresses the special case of degeneracy, a condition that makes the system more difficult to solve. Included are means for detecting degeneracy as well as issues relating to its likelihood in practice. The subsequent part of the paper describes our new procedure, AXIS, which rotates a degenerate problem and then decomposes it into a finite sequence of nondegenerate subproblems of lower dimension. The nondegenerate subproblems are solved using the algorithm DRSALG developed earlier. In the final parts of the paper, the AXIS and DRSALG algorithms are integrated into a single dual response solver termed DR2. DR2 is tested against two nonlinear optimization procedures that have been used frequently in dual response applications. The new solver proves to be extremely effective at locating best-practice operating conditions.     

Hyperspheric Integer Programming

This paper develops an efficient method for finding the optimal solution to linear mathematical programs on 0–1 variables. It is shown that the lattice (0–1) points satisfying some linear constraint of dimension n can equally be represented by those lying in a hypersphere of the same dimension. The lattice points satisfying two linear constraints can be represented by a hypersphere which contains the intersection of the hyperspheres of the two constraints. The method for finding the optimal solution consists of enumerating lattice points which are close to the center of the hypersphere corresponding to the constraints. As soon as a better value of the objective function has been found, than some lower bound, we find a new hypersphere which contains the lattice points of the constraints at which the objective function remains higher than the best known value. We continue in this manner until we have at some stage enumerated all lattice points within a given hypersphere and found none which give a better value.     

A dual decomposition algorithm

Suppose that a large-scale block-diagonal linear programming problem has been solved by the Dantzig—Wolfe decomposition algorithm and that an optimal solution has been attained. Suppose further that it is desired to perform a post-optimality analysis or a complete parametric analysis on the cost-coefficients or the RHS of the linking constraints. Efficient techniques for performing these analyses for the ordinary simplex case have not been easily applied to this case as one operation involves doing a minimizing ratio between all columns of two rows of the tableau. As the columns are not readily known in Dantzig—Wolfe decomposition, other techniques must be used. To date, suggested methods involve solving small linear programs to find these minimizing ratios. In this paper a method is presented which requires solving no linear programs (except possibly in the case of degeneracy of a subproblem) using and utilizing only the information typically stored for Dantzig—Wolfe decomposition.     

A redundancy detection algorithm for fuzzy stochastic multi-objective linear fractional programming problems

The computational complexity of linear and nonlinear programming problems depends on the number of objective functions and constraints involved and solving a large problem often becomes a difficult task. Redundancy detection and elimination provides a suitable tool for reducing this complexity and simplifying a linear or nonlinear programming problem while maintaining the essential properties of the original system. Although a large number of redundancy detection methods have been proposed to simplify linear and nonlinear stochastic programming problems, very little research has been developed for fuzzy stochastic (FS) fractional programming problems. We propose an algorithm that allows to simultaneously detect both redundant objective function(s) and redundant constraint(s) in FS multi-objective linear fractional programming problems. More precisely, our algorithm reduces the number of linear fuzzy fractional objective functions by transforming them in probabilistic–possibilistic constraints characterized by predetermined confidence levels. We present two numerical examples to demonstrate the applicability of the proposed algorithm and exhibit its efficacy.