We use smoothing techniques to solve approximately mildly structured semidefinite programs with many constraints. As smoothing techniques require a specific problem format, we introduce an alternative problem formulation that fulfills the structural assumptions. The resulting algorithm has a complexity that depends linearly both on the number of constraints and on the inverse of the accuracy. … Read more
We consider minimization of nonsmooth functions which can be represented as the composition of a positively homogeneous convex function and a smooth mapping. This is a sufficiently rich class that includes max-functions, largest eigenvalue functions, and norm-1 regularized functions. The bundle method uses an oracle that is able to compute separately the function and subgradient … Read more
In this paper we define semidefinite packing programs and describe an algorithm to approximately solve these problems. Semidefinite packing programs arise in many applications such as semidefinite programming relaxations for combinatorial optimization problems, sparse principal component analysis, and sparse variance unfolding technique for dimension reduction. Our algorithm exploits the structural similarity between semidefinite packing programs … Read more
Proximal bundle methods have been shown to be highly successful optimization methods for unconstrained convex problems with discontinuous first derivatives. This naturally leads to the question of whether proximal variants of bundle methods can be extended to a nonconvex setting. This work proposes an approach based on generating cutting-planes models, not of the objective function … Read more
In this paper a new algorithm to locally minimize nonsmooth, nonconvex functions is developed. We introduce the notion of secants and quasisecants for nonsmooth functions. The quasisecants are applied to find descent directions of locally Lipschitz functions. We design a minimization algorithm which uses quasisecants to find descent directions. We prove that this algorithm converges … Read more
Iterative processes based on Fejer mappings with diminishing problem-specific shifts in the arguments are considered. Such structure allows fine-tuning of Fejer processes by directing them toward selected subsets of attracting sets. Use of various Fejer operators provides ample opportunities for decomposition and parallel computations. Subgradient projection algorithms with sequential and simultaneous projections on segmented constraints … Read more
Gould and Robinson (NAR 08/18, Oxford University Computing Laboratory, 2008) gave global convergence results for a second-derivative SQP method for minimizing the exact $\ell_1$-merit function for a \emph{fixed} value of the penalty parameter. To establish this result, we used the properties of the so-called Cauchy step, which was itself computed from the so-called predictor step. … Read more
An important field of application of non-smooth optimization refers to decomposition of large-scale or complex problems by Lagrangian duality. In this setting, the dual problem consists in maximizing a concave non-smooth function that is defined as the sum of sub-functions. The evaluation of each sub-function requires solving a specific optimization sub-problem, with specific computational complexity. … Read more
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
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