We study the following optimization problem over a dynamical system that consists of several linear subsystems: Given a finite set of n-by-n matrices and an n-dimensional vector, find a sequence of K matrices, each chosen from the given set of matrices, to maximize a convex function over the product of the K matrices and the … Read more
We prove that unless P=NP, there exists no polynomial time (or even pseudo-polynomial time) algorithm that can test whether the optimal value of a nonlinear optimization problem where the objective and constraints are given by low-degree polynomials is attained. If the degrees of these polynomials are fixed, our results along with previously-known “Frank-Wolfe type” theorems … Read more
In a recent paper, it was shown that, for the smooth unconstrained optimization problem, worst-case evaluation complexity $O(\epsilon^{-(p+1)/p})$ may be obtained by means of algorithms that employ sequential approximate minimizations of p-th order Taylor models plus (p + 1)-th order regularization terms. The aforementioned result, which assumes Lipschitz continuity of the p-th partial derivatives, generalizes … Read more
Large-scale constrained convex optimization problems arise in several application domains. First-order methods are good candidates to tackle such problems due to their low iteration complexity and memory requirement. The level-set framework extends the applicability of first-order methods to tackle problems with complicated convex objectives and constraint sets. Current methods based on this framework either rely … Read more
The bootstrap is a nonparametric approach for calculating quantities, such as confidence intervals, directly from data. Since calculating exact bootstrap quantities is believed to be intractable, randomized resampling algorithms are traditionally used. Motivated by the fact that the variability from randomization can lead to inaccurate outputs, we propose a deterministic approach. First, we establish several … Read more
In this survey, we discuss the state-of-the-art of robust combinatorial optimization under uncertain cost functions. We summarize complexity results presented in the literature for various underlying problems, with the aim of pointing out the connections between the different results and approaches, and with a special emphasis on the role of the chosen uncertainty sets. Moreover, … Read more
The unconstrained minimization of a sufficiently smooth objective function $f(x)$ is considered, for which derivatives up to order $p$, $p\geq 2$, are assumed to be available. An adaptive regularization algorithm is proposed that uses Taylor models of the objective of order $p$ and that is guaranteed to find a first- and second-order critical point in … Read more
In the field of robust optimization uncertain data is modeled by uncertainty sets, i.e. sets which contain all relevant outcomes of the uncertain parameters. The complexity of the related robust problem depends strongly on the shape of the uncertainty set. Two popular classes of uncertainty are budgeted uncertainty and ellipsoidal uncertainty. In this paper we … Read more
In this paper, we consider smooth convex optimization problems with simple constraints and inexactness in the oracle information such as value, partial or directional derivatives of the objective function. We introduce a unifying framework, which allows to construct different types of accelerated randomized methods for such problems and to prove convergence rate theorems for them. … Read more
We consider the coordination of planning decisions of a single product in a supply chain composed of one supplier and one retailer by using contracts. We assume that the retailer has the market power to impose his optimal replenishment plan to the supplier. Our concern is on the minimization of the supplier’s cost. In order … Read more