Impulsive Optimal Control of Hybrid Finite-Dimensional Lagrangian Systems

The scope of this dissertation addresses numerical and theoretical issues in the impulsive control of hybrid finite-dimensional Lagrangian systems. In order to treat these aspects, a modeling framework is presented based on the measure-differential inclusion representation of the Lagrangian dynamics. The main advantage of this representation is that it enables the incorporation of set-valued force … Read more

A SECOND DERIVATIVE SQP METHOD WITH IMPOSED DESCENT

Sequential quadratic programming (SQP) methods form a class of highly efficient algorithms for solving nonlinearly constrained optimization problems. Although second derivative information may often be calculated, there is little practical theory that justifies exact-Hessian SQP methods. In particular, the resulting quadratic programming (QP) subproblems are often nonconvex, and thus finding their global solutions may be … Read more

Parallel Space Decomposition of the Mesh Adaptive Direct Search algorithm

This paper describes a parallel space decomposition PSD technique for the mesh adaptive direct search MADS algorithm. MADS extends a generalized pattern search for constrained nonsmooth optimization problems. The objective of the present work is to obtain good solutions to larger problems than the ones typically solved by MADS. The new method PSD-MADS is an … Read more

Primal interior point method for minimization of generalized minimax functions

In this report, we propose a primal interior-point method for large sparse generalized minimax optimization. After a short introduction, where the problem is stated, we introduce the basic equations of the Newton method applied to the KKT conditions and propose a primal interior-point method. Next we describe the basic algorithm and give more details concerning … Read more

Generating set search methods for piecewise smooth problems

We consider a direct search approach for solving nonsmooth minimization problems where the objective function is locally Lipschitz continuous and piecewise continuously differentiable on a finite family of polyhedra. A generating set search method is proposed, which is named “structured” because the structure of the set of nondifferentiability near the current iterate is exploited to … Read more

Gradient methods for minimizing composite objective function

In this paper we analyze several new methods for solving optimization problems with the objective function formed as a sum of two convex terms: one is smooth and given by a black-box oracle, and another is general but simple and its structure is known. Despite to the bad properties of the sum, such problems, both … Read more

Gradient methods for minimizing composite objective function

In this paper we analyze several new methods for solving optimization problems with the objective function formed as a sum of two convex terms: one is smooth and given by a black-box oracle, and another is general but simple and its structure is known. Despite to the bad properties of the sum, such problems, both … Read more

A secant method for nonsmooth optimization

The notion of a secant for locally Lipschitz continuous functions is introduced and a new algorithm to locally minimize nonsmooth, nonconvex functions based on secants is developed. We demonestrate that the secants can be used to design an algorithm to find descent directions of locally Lipschitz continuous functions. This algorithm is applied to design a … Read more

Max-min separability: incremental approach and application to supervised data classification

A new algorithm for the computation of a piecewise linear function separating two finite point sets in $n$-dimensional space is developed and the algorithm is applied to solve supervised data classification problems. The algorithm computes hyperplanes incrementally and it finds as many hyperplanes as necessary to separate two sets with respect to some tolerance. An … Read more

The Speed of Shor’s R-Algorithm

Shor’s r-algorithm is an iterative method for unconstrained optimization, designed for minimizing nonsmooth functions, for which its reported success has been considerable. Although some limited convergence results are known, nothing seems to be known about the algorithm’s rate of convergence, even in the smooth case. We study how the method behaves on convex quadratics, proving … Read more