Semi-infinite programming, duality, discretization and optimality conditions

The aim of this paper is to give a survey of some basic theory of semi-infinite programming. In particular, we discuss various approaches to derivations of duality, discretization and first and second order optimality conditions. Some of the surveyed results are well known while others seem to be less noticed in that area of research. … Read more

A stochastic algorithm for function minimization

Focusing on what an optimization problem may comply with, the so-called convergence conditions have been proposed and sequentially a stochastic optimization algorithm named as DSZ algorithm is presented in order to deal with both unconstrained and constrained optimizations. Its principle is discussed in the theoretical model of DSZ algorithm, from which we present a practical … Read more

Scalable Heuristics for Stochastic Programming with Scenario Selection

We describe computational procedures to solve a wide-ranging class of stochastic programs with chance constraints where the random components of the problem are discretely distributed. Our procedures are based on a combination of Lagrangian relaxation and scenario decomposition, which we solve using a novel variant of Rockafellar and Wets’ progressive hedging algorithm. Experiments demonstrate the … Read more

Numerical Experience with a Recursive Trust-Region Method for Multilevel Nonlinear Optimization

We consider an implementation of the recursive multilevel trust-region algorithm proposed by Gratton, Mouffe, Toint, Weber (2008) for bound-constrained nonlinear problems, and provide numerical experience on multilevel test problems. A suitable choice of the algorithm’s parameters is identified on these problems, yielding a satisfactory compromise between reliability and efficiency. The resulting default algorithm is then … Read more

A new class of large neighborhood path-following interior point algorithms for semidefinite optimization with (\sqrt{n}\log{\frac{{\rm Tr}(X^0S^0)}{\epsilon}})$ iteration complexity

In this paper, we extend the Ai-Zhang direction to the class of semidefinite optimization problems. We define a new wide neighborhood $\N(\tau_1,\tau_2,\eta)$ and, as usual, we utilize symmetric directions by scaling the Newton equation with special matrices. After defining the “positive part” and the “negative part” of a symmetric matrix, we solve the Newton equation … Read more

Convexity in semi-algebraic geometry and polynomial optimization

We review several (and provide new) results on the theory of moments, sums of squares and basic semi-algebraic sets when convexity is present. In particular, we show that under convexity, the hierarchy of semidefinite relaxations for polynomial optimization simplifies and has finite convergence, a highly desirable feature as convex problems are in principle easier to … Read more