In this paper we study the optimal control problem of the heat equation by a distributed control over a subset of the domain, in the presence of a state constraint. The latter is integral over the space and has to be satisfied at each time. Using for the first time the technique of alternative optimality … Read more
In this paper we study the link between a semi-infinite chance-constrained optimization problem and its randomized version, i.e. the problem obtained by sampling a finite number of its constraints. Extending previous results on the feasibility of randomized convex programs, we establish here the feasibility of the solution obtained after the elimination of a portion of … Read more
Lecture notes for the tutorial at the workshop HPOPT 2008 – 10th International Workshop on High Performance Optimization Techniques (Algebraic Structure in Semidefinite Programming), June 11th to 13th, 2008, Tilburg University, The Netherlands. CitationarXiv:0809.2017v1 [math.OC]ArticleDownload View PDF
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
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
We present a line search algorithm for large-scale constrained optimization that is robust and efficient even for problems with (nearly) rank-deficient Jacobian matrices. The method is matrix-free (i.e., it does not require explicit storage or factorizations of derivative matrices), allows for inexact step computations, and is applicable for nonconvex problems. The main components of the … Read more
Dynamic stochastic optimization problems with a large (possibly infinite) number of decision stages and high-dimensional state vector are inherently difficult to solve. In fact, scenario tree based algorithms are unsuitable for problems with many stages, while dynamic programming type techniques are unsuitable for problems with many state variables. This article proposes a stage aggregation scheme … Read more
In this work, we develop duality of the minimum volume circumscribed ellipsoid and the maximum volume inscribed ellipsoid problems. We present a unified treatment of both problems using convex semi–infinite programming. We establish the known duality relationship between the minimum volume circumscribed ellipsoid problem and the optimal experimental design problem in statistics. The duality results … Read more
In many path planning situations we would like to find a path of constrained Euclidean length in 2D that minimises a line integral. We call this the Continuous Length-Constrained Minimum Cost Path Problem (C-LCMCPP). Generally, this will be a non-convex optimization problem, for which continuous approaches only ensure locally optimal solutions. However, network discretisations yield … Read more
We present a multilevel numerical algorithm for the exact solution of the Euclidean trust-region subproblem. This particular subproblem typically arises when optimizing a nonlinear (possibly non-convex) objective function whose variables are discretized continuous functions, in which case the different levels of discretization provide a natural multilevel context. The trust-region problem is considered at the highest … Read more