Using AI methodologies in mathematical modeling of production systems

This research was supported in part by Sandia National Laboratories Grant No. 56-4617. Sandia is managed by AT&;T Technologies for the U.S. Department of Energy.     

Constraint classification in mathematical programming

Consider a set of algebraic inequality constraints defining either an empty or a nonempty feasible region. It is known that each constraint can be classified as either absolutely strongly redundant, relatively strongly redundant, absolutely weakly redundant, relatively weakly redundant, or necessary. We show that is is worth making a distinction between weakly necessary constraints and strongly necessary constraints. We also present afeasible set cover method which can detect both weakly and strongly necessary constraints.The main interest in constraint classification is due to the advantages gained by the removal of redundant constraints. Since classification errors are likely to occur, we examine how the removal of a single constraint can affect the classification of the remaining constraints.     

Ergodicity and symmetric mathematical programs

The paper provides new conditions ensuring the optimality of a symmetric feasible point of certain mathematical programs. It is shown that these conditions generalize and unify most of the known results dealing with optimality of symmetric policies (e.g. [2, 4, 6, 11]). The generalization is based on certain ergodic properties of nonnegative matrices. An application to a socio-economic model dealing with optimization of a welfare function is presented.     

Constraint augmentation in pseudo-singularly perturbed linear programs

In this paper we study a linear programming problem with a linear perturbation introduced through a parameter ε > 0. We identify and analyze an unusual asymptotic phenomenon in such a linear program. Namely, discontinuous limiting behavior of the optimal objective function value of such a linear program may occur even when the rank of the coefficient matrix of the constraints is unchanged by the perturbation. We show that, under mild conditions, this phenomenon is a result of the classical Slater constraint qualification being violated at the limit and propose an iterative, constraint augmentation approach for resolving this problem.     

Existences theorems of systems of vector quasi-equillibrium problems and mathematical programs with equilibrium constraint

In this paper, we introduce systems of vector quasi-equilibrium problems and prove the existence of their solutions. As applications of our results, we derive the existence theorems for solution of system of vector quasi-saddle point problem, the existences theorems of a solution of system of generalized quasi-minimax inequalities, the mathematical program with equilibrium constraint, semi-infinite and bilevel problems.     

Pseudo-duality in mathematical programs with quotients and products

A new approach to duality in fractional programs is described. The techniques used are based on the theory of pseudoduality. The results are generalized to include mathematical programs with quotients and products of finitely many functionals. The duals of the linear and quadratic cases are explicitly calculated, demonstrating the power of pseudo-duality theory.The author wishes to thank Prof. S. Schaible for his helpful comments. In discussions with Prof. Schaible, it was found that the duals for the linear and the quadratic cases are identical with those derived in Ref. 3.     

Synergy of homomorphisms in relational systems

Suppose that a relational system $\text{R}$ can be exhausted (in an obvious sense) by a family R_ω, $\text{ω ? Ω}$, of subrelational systems, each of which can be mapped by a homomorphism onto a subrelational system S_ω of a second relational system $\text{S}$. We show that, under suitable finiteness conditions, there is a homomorphism from $\text{R}$ into $\text{S}$ which finitely agrees with the homomorphisms mapping R_ω onto S_ω. A similar result holds for isomorphisms.     

Constraint identification and algorithm stabilization for degenerate nonlinear programs

 In the vicinity of a solution of a nonlinear programming problem at which both strict complementarity and linear independence of the active constraints may fail to hold, we describe a technique for distinguishing weakly active from strongly active constraints. We show that this information can be used to modify the sequential quadratic programming algorithm so that it exhibits superlinear convergence to the solution under assumptions weaker than those made in previous analyses. Received: December 18, 2000 / Accepted: January 14, 2002 Published online: September 27, 2002 RID="★" ID="★" Research supported by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Advanced Scientific Computing Research, U.S. Department of Energy, under Contract W-31-109-Eng-38. Key words. nonlinear programming problems – degeneracy – active constraint identification – sequential quadratic programming     

Stochastic mathematical programs with equilibrium constraints,modelling and sample average approximation

In this article, we discuss the sample average approximation (SAA) method applied to a class of stochastic mathematical programs with variational (equilibrium) constraints. To this end, we briefly investigate the structure of both–the lower level equilibrium solution and objective integrand. We show almost sure convergence of optimal values, optimal solutions (both local and global) and generalized Karush–Kuhn–Tucker points of the SAA program to their true counterparts. We also study uniform exponential convergence of the sample average approximations, and as a consequence derive estimates of the sample size required to solve the true problem with a given accuracy. Finally, we present some preliminary numerical test results.     

Characterization of perturbed mathematical programs and interval analysis

Several authors have used interval arithmetic to deal with parametric or sensitivity analysis in mathematical programming problems. Several reported computational experiments have shown how interval arithmetic can provide such results. However, there has not been a characterization of the resulting solution interval in terms of the usual sensitivity analysis results. This paper presents a characterization of perturbed convex programs and the resulting solution intervals.Interval arithmetic was developed as a mechanism for dealing with the inherent error associated with numerical computations using a computational device. Here it is used to describe error in the parameters. We show that, for convex programs, the resulting solution intervals can be characterized in terms of the usual sensitivity analysis results. It has been often reported in the literature that even well behaved convex problems can exhibit pathological behavior in the presence of data perturbations. This paper uses interval arithmetic to deal with such problems, and to characterize the behavior of the perturbed problem in the resulting interval. These results form the foundation for future computational studies using interval arithmetic to do nonlinear parametric analysis.     

Constraint qualifications in a class of nondifferentiable mathematical programming problems

The Kuhn-Tucker type necessary optimality conditions are given for the problem of minimizing the sum of a differentiable function and a convex function subject to a set of differentiable nonlinear inequalities on a convex subset C of , under the conditions similar to the Kuhn-Tucker constraint qualification or the Arrow-Hurwicz-Uzawa constraint qualification. The case when the set C is open (not necessarily convex) is shown to be a special one of our results, which helps us to improve some of the existing results in the literature.     

An identification algorithm in fuzzy relational systems

The paper presents an effective identification method in fuzzy relational systems. We propose an algorithm for constructing models on the basis of fuzzy and nonfuzzy data with the aid of fuzzy discretization and clustering techniques. The usefulness of the method provided is demonstrated by means of two numerical examples. Also a possible way of generating a linguistic decision-making algorithm is discussed.     

On powers of relational and algebraic systems

We extend the well-known Birkhoff’s operation of cardinal power from partially ordered sets onto n-ary relational systems. The extended power is then studied not only for n-ary relational systems but also for some of their special cases, namely partial algebras and total algebras. It turns out that a concept of diagonality plays an important role when studying the powers.     

Constraint qualification in a general class of Lipschitzian mathematical programming problems

The Kuhn–Tucker type necessary optimality conditions are given for the problem of minimizing the sum of a differentiable function and a locally Lipschitzian function subject to a set of differentiable nonlinear inequalities on a convex subset C of , under the condition of a generalized Kuhn–Tucker constraint qualification or a generalized Arrow–Hurwicz–Uzawa constraint qualification. The case when the set C is open is shown to be a special one of our results, which helps us to improve some of the existing results in the literature. To finish we consider several test problems.