The issue of characterizing completely efficient (Pareto) solutions to a fractional vector (multiobjective or multicriteria) minimization problem, where the involved functions are convex, has not been addressed previously. Thanks to an earlier characterization of weak efficiency in difference vector optimization by El Maghri, we get a vectorial necessary and sufficient condition given in terms of … Read more
In this paper, we introduce a new effective matrix adaptation evolution strategy (MADFO) for noisy derivative-free optimization problems. Like every MAES solver, MADFO consists of three phases: mutation, selection and recombination. MADFO improves the mutation phase by generating good step sizes, neither too small nor too large, that increase the probability of selecting mutation points … Read more
\(\) We study the equivalence of several well-known sufficient optimality conditions for a general quadratically constrained quadratic program (QCQP). The conditions are classified in two categories. The first one is for determining an optimal solution and the second one is for finding an optimal value. The first category of conditions includes the existence of a … Read more
We study the convergence of random function iterations for finding an invariant measure of the corresponding Markov operator. We call the problem of finding such an invariant mea- sure the stochastic fixed point problem. This generalizes earlier work studying the stochastic feasibility problem, namely, to find points that are, with probability 1, fixed points of … Read more
We present a novel response surface method for global optimisation of an expensive and noisy (black-box) objective function, where error bounds on the deviation of the observed noisy function values from their true counterparts are available. The method is based on the well-established RBF method by Gutmann (2001a,c) for minimising an expensive and deterministic objective … Read more
Quadratic programs with box constraints involve minimizing a possibly nonconvex quadratic function subject to lower and upper bounds on each variable. This is a well-known NP-hard problem that frequently arises in various applications. We focus on two convex relaxations, namely the RLT (Reformulation-Linearization Technique) relaxation and the SDP-RLT relaxation obtained by adding semidefinite constraints to … Read more
\(\) Globally optimizing a nonconvex quadratic over the intersection of $m$ balls in $\mathbb{R}^n$ is known to be polynomial-time solvable for fixed $m$. Moreover, when $m=1$, the standard semidefinite relaxation is exact. When $m=2$, it has been shown recently that an exact relaxation can be constructed using a disjunctive semidefinite formulation based essentially on two … Read more
We use sensitivity analysis to design optimality-based discretization (cutting-plane) methods for the global optimization of nonconvex semi-infinite programs (SIPs). We begin by formulating the optimal discretization of SIPs as a max-min problem and propose variants that are more computationally tractable. We then use parametric sensitivity theory to design an efficient method for solving these max-min … Read more
In low-rank matrix recovery, the goal is to recover a low-rank matrix, given a limited number of linear and possibly noisy measurements. Low-rank matrix recovery is typically solved via a nonconvex method called Burer-Monteiro factorization (BM). If the rank of the ground truth is known, BM is free of sub-optimal local solutions, and its true solutions … Read more
We study optimization methods applied to minimizing forces for poses and movements of chained Stewart platforms (SPs) that we call an “Assembler” Robot. These chained SPs are parallel mechanisms that are stronger, stiffer, and more precise, on average, than their serial counterparts at the cost of a smaller range of motion. Linking these units in … Read more