A mathematical programming problem with singular mixed pointwise-integral constraints

We investigate an infinite dimensional optimization problem which constraints are singular integral-pointwise ones. We give some partial results of existence for a solution in some particular cases. However, the lack of compactness, even in L¹ prevents to conclude in the general case. We give an existence result for a weak solution (as a measure) that we are able to describe. The regularity of such a solution is still an open problem.     

Iterative algorithm for mathematical programming problems with preconvex constraints

An iterative algorithm is proposed for minimizing a convex function on a set defined as the set-theoretic difference between a convex set and the union of several convex sets. The convergence of the algorithm is proved in terms of necessary conditions for a local minimum.     

Necessary optimality conditions for minimax programming problems with mathematical constraints

In this paper, necessary optimality conditions in terms of upper and/or lower subdifferentials of both cost and constraint functions are derived for minimax optimization problems with inequality, equality and geometric constraints in the setting of non-differentiatiable and non-Lipschitz functions in Asplund spaces. Necessary optimality conditions in the fuzzy form are also presented. An application of the fuzzy necessary optimality condition is shown by considering minimax fractional programming problem.     

On the convergence properties of modified augmented Lagrangian methods for mathematical programming with complementarity constraints

In this paper, we present new convergence results of augmented Lagrangian methods for mathematical programs with complementarity constraints (MPCC). Modified augmented Lagrangian methods based on four different algorithmic strategies are considered for the constrained nonconvex optimization reformulation of MPCC. We show that the convergence to a global optimal solution of the problem can be ensured without requiring the boundedness condition of the multipliers.     

The method of feasible directions for mathematical programming problems with preconvex constraints

The convergence of the method of feasible directions is proved for the case of the smooth objective function and a constraint in the form of the difference of convex sets (the so-called preconvex set). It is shown that the method converges to the set of stationary points, which generally is narrower than the corresponding set in the case of a smooth function and smooth constraints. The scheme of the proof is similar to that proposed earlier by Karmanov.     

On duality for mathematical programs with vanishing constraints

We make a review of several variants of ergodicity for continuous-time Markov chains on a countable state space. These include strong ergodicity, ergodicity in weighted-norm spaces, exponential and subexponential ergodicity. We also study uniform exponential ergodicity for continuous-time controlled Markov chains, as a tool to deal with average reward and related optimality criteria. A discussion on the corresponding ergodicity properties is made, and an application to a controlled population system is shown.     

A semismooth sequential quadratic programming method for lifted mathematical programs with vanishing constraints

Mathematical programs with vanishing constraints are a difficult class of optimization problems with important applications to optimal topology design problems of mechanical structures. Recently, they have attracted increasingly more attention of experts. The basic difficulty in the analysis and numerical solution of such problems is that their constraints are usually nonregular at the solution. In this paper, a new approach to the numerical solution of these problems is proposed. It is based on their reduction to the so-called lifted mathematical programs with conventional equality and inequality constraints. Special versions of the sequential quadratic programming method are proposed for solving lifted problems. Preliminary numerical results indicate the competitiveness of this approach.     

Pseudo duality in mathematical programming: Unconstrained problems and problems with equality constraints

In this work non-convex programs are analyzed via Legendre transform. The first part includes definitions and the classification of programs that can be handled by the transformation. It is shown that differentiable functions that are represented as a sum of strictly concave and convex functions belong to this class. Conditions under which a function may have such representation are given. Pseudo duality is defined and the pseudo duality theorem for non linear programs with equality constraints is proved.The techniques described are constructive ones, and they enable tocalculate explicitly a pseudo dual once the primal program is given. Several examples are included. In the convex case these techniques enable the explicit calculation of the dual even in cases where direct calculation was not possible.     

A method of multipliers for mathematical programming problems with equality and inequality constraints

This paper deals with the numerical solution of the general mathematical programming problem of minimizing a scalar functionf(x) subject to the vector constraints φ(x)=0 and ψ(x)≥0. The approach used is an extension of the Hestenes method of multipliers, which deals with the equality constraints only. The above problem is replaced by a sequence of problems of minimizing the augmented penalty function Ω(x, λ, μ,k)=f(x)+λ ^T φ(x)+kφ ^T (x)φ(x) ?μ ^T \(\tilde \psi \) (x)+k \(\tilde \psi \) ^T (x) \(\tilde \psi \) (x). The vectors λ and μ, μ ≥ 0, are respectively the Lagrange multipliers for φ(x) and \(\tilde \psi \) (x), and the elements of \(\tilde \psi \) (x) are defined by \(\tilde \psi \) _(j)(x)=min[ψ_(j)(x), (1/2k) μ_(j)]. The scalark>0 is the penalty constant, held fixed throughout the algorithm. Rules are given for updating the multipliers for each minimization cycle. Justification is given for trusting that the sequence of minimizing points will converge to the solution point of the original problem.     

On abstract duality in mathematical programming

It is shown that duality in mathematical programming can be treated as a purely order theoretic concept which leads to some applications in economics. Conditions for strong duality results are given. Furthermore the underlying sets are endowed with (semi-)linear structures, and the perturbation function of arising linear and integer problems, which include bottleneck problems and extremal problems (in the sense of K. Zimmermann), is investigated.

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Zusammenfassung In dieser Arbeit wird aufgezeigt, daß Dualitätskonzepte der mathematischen Optimierung in ordnungstheoretischem Rahmen beschrieben werden können. Dies führt u.a. auf neue Anwendungen in der Ökonomie. Ferner werden Bedingungen hergeleitet, unter denen starke Dualitätsaussagen gelten. Sodann werden die zugrundeliegenden Mengen mit algebraischen Strukturen versehen und es werden Dualitätssätze für lineare und ganzzahlige Programme über diesen Mengen bewiesen. Darunter fallen nicht nur die klassischen linearen und ganzzahligen Programme, sondern auch Probleme mit Engpaßzielfunktion und extremale Probleme im Sinne von K. Zimmermann.

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This paper was partially supported by the NATO Research Grants Programme under SRG 8.     

On M-stationary points for mathematical programs with equilibrium constraints

Mathematical programs with equilibrium constraints are optimization problems which violate most of the standard constraint qualifications. Hence the usual Karush-Kuhn-Tucker conditions cannot be viewed as first order optimality conditions unless relatively strong assumptions are satisfied. This observation has lead to a number of weaker first order conditions, with M-stationarity being the strongest among these weaker conditions. Here we show that M-stationarity is a first order optimality condition under a very weak Abadie-type constraint qualification. Our approach is inspired by the methodology employed by Jane Ye, who proved the same result using results from optimization problems with variational inequality constraints. In the course of our investigation, several concepts are translated to an MPEC setting, yielding in particular a very strong exact penalization result.     

On symmetric duality in nondifferentiable mathematical programming with F-convexity

Usual symmetric duality results are proved for Wolfe and Mond-Weir type nondifferentiable nonlinear symmetric dual programs under F-convexity F-concavity and F-pseudoconvexity F-pseudoconcavity assumptions. These duality results are then used to formulate Wolfe and Mond-Weir type nondifferentiable minimax mixed integer dual programs and symmetric duality theorems are established. Moreover, nondifferentiable fractional symmetric dual programs are studied by using the above programs.     

Solving stochastic mathematical programs with equilibrium constraints via approximation and smoothing implicit programming with penalization

In this paper, we consider the stochastic mathematical programs with linear complementarity constraints, which include two kinds of models called here-and-now and lower-level wait-and-see problems. We present a combined smoothing implicit programming and penalty method for the problems with a finite sample space. Then, we suggest a quasi-Monte Carlo approximation method for solving a problem with continuous random variables. A comprehensive convergence theory is included as well. We further report numerical results with the so-called picnic vender decision problem.     

On implicit function theorems for set-valued maps and their application to mathematical programming under inclusion constraints

In this paper we establish some implicit function theorems for a class of locally Lipschitz set-valued maps and then apply them to investigate some questions concerning the stability of optimization problems with inclusion constraints. In consequence we have an extension of some of the corresponding results of Robinson, Aubin, and others.     

On sufficient optimality conditions for mathematical programs with equaiity constraints

A number of sufficiency theorems in the mathematical programming literature, concerning problems with equality constraints, are shown to be trivial consequences of the corresponding results for inequality constraints.This work was supported by NSF Grant No. ECS-8214081. Research by the first author was done while a visitor at La Trobe University.     

On the cones of tangents with applications to mathematical programming

In this study, we present a unifying framework for the cones of tangents to an arbitrary set and some of its applications. We highlight the significance of these cones and their polars both from the point of view of differentiability and subdifferentiability theory and the point of view of mathematical programming. This leads to a generalized definition of a subgradient which extends the well-known definition of a subgradient of a convex function to the nonconvex case. As an application, we develop necessary optimality conditions for a min-max problem and show that these conditions are also sufficient under moderate convexity assumptions.     

On alternative optimal solutions to quasimonotonic programming with linear constraints

In this paper, the nonlinear programming problem with quasimonotonic （ both quasiconvex and quasiconcave ）objective function and linear constraints is considered. With the decomposition theorem of polyhedral sets, the structure of optimal solution set for the programming problem is depicted. Based on a simplified version of the convex simplex method, the uniqueness condition of optimal solution and the computational procedures to determine all optimal solutions are given, if the uniqueness condition is not satisfied. An illustrative example is also presented.     

Elementary proof of classical necessary optimality conditions for a mathematical programming problem with inequality constraints

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