The projected subgradient method for constrained minimization repeatedly interlaces subgradient steps for the objective function with projections onto the feasible region, which is the intersection of closed and convex constraints sets, to regain feasibility. The latter poses a computational difficulty and, therefore, the projected subgradient method is applicable only when the feasible region is “simple … Read more
In the context of sequential methods for solving general nonlinear programming problems, it is usual to work with augmented subproblems instead of the original ones, tackled by the $\ell_1$-penalty function together with the shortcut usage of a convenient penalty parameter. This paper addresses the theoretical reasoning behind handling the original subproblems by such an augmentation … Read more
For a long time, the bilevel programming problem has essentially been considered as a special case of mathematical programs with equilibrium constraints (MPECs), in particular when the so-called KKT reformulation is in question. Recently though, this widespread believe was shown to be false in general. In this paper, other aspects of the difference between both … Read more
In this report, we state and prove first- and second-order necessary conditions in Pontryagin form for optimal control problems with pure state and mixed control-state constraints. We say that a Lagrange multiplier of an optimal control problem is a Pontryagin multiplier if it is such that Pontryagin’s minimum principle holds, and we call optimality conditions … Read more
In this report, given a reference feasible trajectory of an optimal control problem, we say that the quadratic growth property for bounded strong solutions holds if the cost function of the problem has a quadratic growth over the set of feasible trajectories with a bounded control and with a state variable sufficiently close to the … Read more
We consider the problem of identifying the densest k-node subgraph in a given graph. We write this problem as an instance of rank-constrained cardinality minimization and then relax using the nuclear and l1 norms. Although the original combinatorial problem is NP-hard, we show that the densest k-subgraph can be recovered from the solution of our … Read more
In this paper, we propose new linesearch-based methods for nonsmooth optimization problems when first-order information on the problem functions is not available. In the first part, we describe a general framework for bound-constrained problems and analyze its convergence towards stationary points, using the Clarke-Jahn directional derivative. In the second part, we consider inequality constrained optimization … Read more
We present a filter linesearch method for solving general nonlinear and nonconvex optimization problems. The method is of the filter variety, but uses a robust (always feasible) subproblem based on an exact penalty function to compute a search direction. This contrasts traditional filter methods that use a (separate) restoration phase designed to reduce infeasibility until … Read more
This paper is a continuation of our previous paper were we presented generalizations of the Dennis-Mor\’e theorem to characterize q-superliner convergences of quasi-Newton methods for solving equations and variational inequalities in Banach spaces. Here we prove Dennis-Mor\’e type theorems for inexact quasi-Newton methods applied to variational inequalities in finite dimensions. We first consider variational inequalities … Read more
This paper presents four optimization models for solving nonlinear equation systems. The models accommodate both over-specified and under-specified systems. A variable endogenization technique that improves efficiency is introduced, and a basic comparative study shows one of the methods presented to be very effective. Citation Siwale, I. (2013). Solution of nonlinear equation systems via optimization. Technical … Read more