K e y w o r d s M u l t i o b j e c t i v e optimization, Mangasarian-l'~romovitz type conditions, Second-order optimality conditions. 1. I N T R O D U C T I O N We consider the following constrained multiobjective program: min f(x), subject to x E X, (1) int R~ where the feasible region is described by inequalities and equalities X := {x • R": g(x) <_ O, h(x) = 0}, with f : R" ---* R t, g : R ...

Multistage stochastic optimization leads to NLPs over scenario trees that become extremely large when many time stages or fine discretizations of the probability space are required. Interior-point methods are well suited for these problems if the arising huge, structured KKT systems can be solved efficiently, for instance, with a large scenario tree but a moderate number of variables per node. ...

The bilevel program is a sequence of two optimization problems where the constraint region of the upper level problem is determined implicitly by the solution set to the lower level problem. The classical approach to solving such a problem is to replace the lower level problem by its Karush–Kuhn–Tucker (KKT) condition and solve the resulting mathematical programming problem with equilibrium con...

This paper presents a branch-and-bound algorithm for nonconvex quadratic programming, which is based on solving semidefinite relaxations at each node of the enumeration tree. The method is motivated by a recent branch-and-cut approach for the box-constrained case that employs linear relaxations of the first-order KKT conditions. We discuss certain limitations of linear relaxations when handling...

Optimization problems constrained by nonlinear partial differential equations have been the focus of intense research in scientific computing lately. Current methods for the parallel numerical solution of such problems involve sequential quadratic programming (SQP), with either reduced or full space approaches. In this paper we propose and investigate a class of parallel full space SQP Lagrange...

Multiple Modular Design (MMD) is the method of designing a set of standard modules to meet demands for di erent functions. This work aims to nd an optimal set of modules given the demands. In this paper, we start with Evans' nonlinear programming model of MMD. By exploring the special structure of the formulation, we develop several properties of optimal solutions. With these properties, we dev...

In this paper, an augmented Lagrangian proximal alternating (ALPA) method is proposed for two class of large-scale sparse discrete constrained optimization problems in which a sequence of augmented Lagrangian subproblems are solved by utilizing proximal alternating linearized minimization framework and sparse projection techniques. Under the MangasarianFromovitz and the basic constraint qualifi...

It is known that the Karush-Kuhn-Tucker (KKT) conditions of semidefinite programming can be reformulated as a nonsmooth system via the metric projector over the cone of symmetric and positive semidefinite matrices. We show in this paper that the primal and dual constraint nondegeneracies, the strong regularity, the nonsingularity of the B-subdifferential of this nonsmooth system, and the nonsin...

This chapter is mainly about William Karush and his role in the Karush-KuhnTucker theorem of nonlinear programming. It tells the story of fundamental optimization results that he obtained in his master’s thesis: results that he neither published nor advertised and that were later independently rediscovered and published by Harold W. Kuhn and Albert W. Tucker. The principal result – which concer...