On solving convex quadratic semi-infinite programming probelms

In this paper we study a from of convex quadratic semi-infinite programming problems with finitely many variables and infinitely many constraints over a compact metric space. An entropic path-following algorithum is introduced with a convergence proof. Some practical implementations and numerical experiments are also included     

The almost chebyshev property in linear semi-infinite programming

We consider a linear semi-infinite programming problem where the index set of the constraints is compact and the constraint functions are continuous on it. The set of all continuous functions on this index set as right hand sides are the parameter set. We investigate how large various unicity sets are.We state a condition on the objective function vector and the “matrix” of the problem which characterizes when the set of a parameters with a non-unique optimal point is a set of the first Baire category in the solvability set. This is the case if and only if the unicity set is a dense subset of the solvability set. Under the same assumptions it is even true that the interior of the strong unicity set is I also dense. If the index set of the constraints contains a dense subset with the property that each point1 is a G ₈-set, then the parameters of the strong unicity set, such that the optimal point satisfies the linear independence constraint qualification, are also dense.

We apply our results to a characterization of a unique continuous selection for the optimal set I mapping and to a one-sided L ₁-approximation problem     

Dimension and finite reduction in linear semi-infinite programming

In the first part, different characterizations for the dimension of the feasible set in linear semi-infinite programming are provided. They involve the corresponding dimensions of some parameter sets, as the consequent inequalities cone and its lineality subspace. The remaining sections of the paper deal with Farkas–Minkowski systems. The third section is devoted to establish some results concerning the optimal set and its dimension, exploiting its strong relation with a particular parameter cone

associated with the corresponding unstable constraints. The last section approaches the finite reducibility problem. We have intended to characterize those finite subproblems with the same optimal value as the original problem, by means of a simplc dual analysis, based on the main results derived before.     

An unconstrained convex programming approach to solving convex quadratic programming problems

In this paper, we derive an unconstrained convex programming approach to solving convex quadratic programming problems in standard form. Related duality theory is established by using two simple inequalities. An ?-optimal solution is obtained by solving an unconstrained dual convex program. A dual-to-primal conversion formula is also provided. Some preliminary computational results of using a curved search method is included     

Generalized semi-infinite programming: numerical aspects

Generalized semi-infinite optimization problems (GSIP) are considered. It is investigated how the numerical methods for standard semi-infinite programming (SIP) can be extended to GSIP. Newton methods can be extended immediately. For discretization methods the situation is more complicated. These difficulties are discussed and convergence results for a discretization- and an exchange method are derived under fairly general assumptions on GSIP. The question is answered under which conditions GSIP represents a convex problem.     

An unconstrained convex programming view of linear programming

The major interest of this paper is to show that, at least in theory, a pair of primal and dual -optimal solutions to a general linear program in Karmarkar's standard form can be obtained by solving an unconstrained convex program. Hence unconstrained convex optimization methods are suggested to be carefully reviewed for this purpose.     

On the entropic regularization method for solving min-max problems with applications

Consider a min-max problem in the form of min _xX max_1im {f _i (x)}. It is well-known that the non-differentiability of the max functionF(x) max_1im {f _i (x)} presents difficulty in finding an optimal solution. An entropic regularization procedure provides a smooth approximationF _p(x) that uniformly converges toF(x) overX with a difference bounded by ln(m)/p, forp > 0. In this way, withp being sufficiently large, minimizing the smooth functionF _p(x) overX provides a very accurate solution to the min-max problem. The same procedure can be applied to solve systems of inequalities, linear programming problems, and constrained min-max problems.This research work was supported in part by the 1995 NCSC-Cray Research Grant and the National Textile Center Research Grant S95-2.     

Linear programming in measure spaces

This paper studies a linear programming problem in measure spaces (LPM). Several results are obtained. First, the optimal value of LPM can be equal to the optimal value of the dual problem (DLPM), but the solution of DLPM may be not exist in its feasible region. Sccond, :he relations between the optimal solution of LPM and the extreme point of the feasible region of LPM are discussed. In order to investigate the conditions under which a feasible solution becomes an extremal point, the inequality constraint of LPM is transformed to an equality constraint. Third, the LPM can be reformulated to be a general capacity problem (GCAP) or a linear semi-infinite programming problem (LSIP = SIP), and under appropriate restrictioiis, the algorithm developed by the authors in [7] and [8] are applicable for developing an approximation scheme for the optimal solution of LPM     

A geometric approach for solving fuzzy linear programming problems

In this paper we first recall some definitions and results of fuzzy plane geometry, and then introduce some definitions in the geometry of two-dimensional fuzzy linear programming (FLP). After defining the optimal solution based on these definitions, we use the geometric approach for obtaining optimal solution(s) and show that the algebraic solutions obtained by Zimmermann method (ZM) and our geometric solutions are the same. Finally, numerical examples are solved by these two methods.     

Optimality conditions for nondifferentiable convex semi-infinite programming

This paper gives characterizations of optimal solutions to the nondifferentiable convex semi-infinite programming problem, which involve the notion of Lagrangian saddlepoint. With the aim of giving the necessary conditions for optimality, local and global constraint qualifications are established. These constraint qualifications are based on the property of Farkas-Minkowski, which plays an important role in relation to certain systems obtained by linearizing the feasible set. It is proved that Slater's qualification implies those qualifications.     

An iterative approach to solve multiobjective linear fractional programming problems

This paper suggests an iterative parametric approach for solving multiobjective linear fractional programming (MOLFP) problems which only uses linear programming to obtain efficient solutions and always converges to an efficient solution. A numerical example shows that this approach performs better than some existing algorithms. Randomly generated MOLFP problems are also solved to demonstrate the performance of new introduced algorithm.     

A projected lagrangian algorithm for semi-infinite programming

A globally convergent algorithm is presented for the solution of a wide class of semi-infinite programming problems. The method is based on the solution of a sequence of equality constrained quadratic programming problems, and usually has a second order convergence rate. Numerical results illustrating the method are given.     

Strong duality for inexact linear programming

In this paper, we apply the Dubovitskii-Milyutin approach to derive strong duality theorems for inexact linear programming problems. Inexact linear programming deals with the standard linear problem in which the data is not well known and it is supposed to lie in certain given convex sets. The case of parametric dependence of the data is particularly analyzed and relations with semi-infinite and

semi-definite programming are also commented.     

Path-following proximal approach for solving Ill-posed convex semi-infinite programming problems

For a class of ill-posed, convex semi-infinite programming problems, a regularized path-following strategy is developed. This approach consists in a coordinated application of adaptive discretization and prox-regularization procedures combined with a penalty method. At each iteration, only an approximate minimum of a strongly convex differentiable function has to be calculated, and this can be done by any fast-convergent algorithm. The use of prox-regularization ensures the convergence of the iterates to some solution of the original problem. Due to regularization, an efficient deleting rule is applicable, which excludes an essential part of the constraints in the discretized problems.This research was supported by the German Research Society (DFG).The authors are grateful to the anonymous referees for their valuable comments.     

Connections between generalized,inexact and semi-infinite linear programming

This paper presents duality results between generalized and inexact linear programs and describes a special type of linear semi-infinite programs in connection with the programs above mentioned. In order to solve inexact linear programs a corresponding auxiliary problem can be formulated which is explicitly solvable. However, this auxiliary problem is a reformulation of the reduced semi-infinite problem. Therefore, all the numerical methods for solving semi-infinite linear programs can be used for the numerical treatment of inexact and generalized linear programs.

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Zusammenfassung Die vorliegende Arbeit zeigt Dualitätsergebnisse zwischen verallgemeinerten und inexakten linearen Programmen auf und beschreibt einen speziellen Typ linear-semi-infiniter Programme in Zusammenhang mit den oben erwähnten Optimierungsaufgaben. Um inexakte lineare Programme zu lösen wird ein Hilfsproblem aufgestellt, das explizit lösbar ist. Dieses Hilfsproblem ist eine Reformulierung des reduzierten semi-infiniten Problems. Daher können alle numerischen Methoden zur Lösung semi-infiniter linearer Programme auch zur numerischen Behandlung von inexakten und verallgemeinerten lineraren Programmen herangezogen werden.

     

Solving nonlinear programming problems with very many constraints

For solving the smooth constrained nonlinear programming problem, sequential quadratic programming (SQP) methods are considered to be the standard tool, as long as they are applicable. However one possible situation preventing the successful solution by a standard SQP-technique, arises if problems with a very large number of constraints are to be solved. Typical applications are semi-infinite or min-max optimization, optimal control or mechanical structural optimization. The proposed technique proceeds from a user defined number of linearized constraints, that is to be used internally to determine the size of the quadratic programming subproblem. Significant constraints are then selected automatically by the algorithm. Details of the numerical implementation and some experimental results are presented     

LP relaxations for a class of linear semi-infinite programming problems

In this paper, we consider a subclass of linear semi-infinite programming problems whose constraint functions are polynomials in parameters and index sets are polyhedra. Based on Handelman’s representation of positive polynomials on a polyhedron, we propose two hierarchies of LP relaxations of the considered problem which respectively provide two sequences of upper and lower bounds of the optimum. These bounds converge to the optimum under some mild assumptions. Sparsity in the LP relaxations is explored for saving computational time and avoiding numerical ill behaviors.     

An implementation of the simplex method for linear programming problems with variable upper bounds

Special methods for dealing with constraints of the formx _j x _k , called variable upper bounds, were introduced by Schrage. Here we describe a method that circumvents the massive degeneracy inherent in these constraints and show how it can be implemented using triangular basis factorizations.This research was partially supported by National Science Foundation Grant ECS-7921279 and by a Guggenheim fellowship.     

A hybrid approach to discrete mathematical programming

The dynamic programming and branch-and-bound approaches are combined to produce a hybrid algorithm for separable discrete mathematical programs. Linear programming is used in a novel way to compute bounds. Every simplex pivot permits a bounding test to be made on every active node in the search tree. Computational experience is reported.     

An extension of the simplex algorithm for semi-infinite linear programming

We present a primal method for the solution of the semi-infinite linear programming problem with constraint index setS. We begin with a detailed treatment of the case whenS is a closed line interval in . A characterization of the extreme points of the feasible set is given, together with a purification algorithm which constructs an extreme point from any initial feasible solution. The set of points inS where the constraints are active is crucial to the development we give. In the non-degenerate case, the descent step for the new algorithm takes one of two forms: either an active point is dropped, or an active point is perturbed to the left or right. We also discuss the form of the algorithm when the extreme point solution is degenerate, and in the general case when the constraint index set lies in ^p . The method has associated with it some numerical difficulties which are at present unresolved. Hence it is primarily of interest in the theoretical context of infinite-dimensional extensions of the simplex algorithm.