We consider the sensitivity of semidefinite programs (SDPs) under perturbations. It is well known that the optimal value changes continuously under perturbations on the right hand side in the case where the Slater condition holds in the primal problems. In this manuscript, we observe by investigating a concrete SDP that the optimal value can be … Read more
We consider the gradient (or steepest) descent method with exact line search applied to a strongly convex function with Lipschitz continuous gradient. We establish the exact worst-case rate of convergence of this scheme, and show that this worst-case behavior is exhibited by a certain convex quadratic function. We also extend the result to a noisy … Read more
We provide a framework for computing the exact worst-case performance of any algorithm belonging to a broad class of oracle-based first-order methods for composite convex optimization, including those performing explicit, projected, proximal, conditional and inexact (sub)gradient steps. We simultaneously obtain tight worst-case guarantees and explicit instances of optimization problems on which the algorithm reaches this … Read more
Growth-optimal portfolios are guaranteed to accumulate higher wealth than any other investment strategy in the long run. However, they tend to be risky in the short term. For serially uncorrelated markets, similar portfolios with more robust guarantees have been recently proposed. This paper extends these robust portfolios by accommodating non-zero autocorrelations that may reflect investors’ … Read more
We consider semidefinite optimization problems that include constraints that G(x) and H(x) are positive semidefinite (PSD), where the components of the symmetric matrices G(x) and H(x) are affine functions of an n-vector x. In such a case we obtain a new constraint that a matrix K(x,X) is PSD, where the components of K(x,X) are affine … Read more
Distributionally robust optimization (DRO) is widely used, because it offers a way to overcome the conservativeness of robust optimization without requiring the specificity of stochastic optimization. On the computational side, many practical DRO instances can be equivalently (or approximately) formulated as semidefinite programming (SDP) problems via conic duality of the moment problem. However, despite being … Read more
The bounded degree sum-of-squares (BSOS) hierarchy of Lasserre, Toh, and Yang [EURO J. Comput. Optim., 2015] constructs lower bounds for a general polynomial optimization problem with compact feasible set, by solving a sequence of semi-definite programming (SDP) problems. Lasserre, Toh, and Yang prove that these lower bounds converge to the optimal value of the original … Read more
In this paper we analyze general two-term disjunctions on a regular cone $\K$ and derive a general form for a family of convex inequalities which are valid for the resulting nonconvex sets. Under mild technical assumptions, these inequalities collectively describe the closed convex hulls of these disjunctions, and if additional conditions are satisfied, a single … Read more
We consider the maximum k-cut problem that consists in partitioning the vertex set of a graph into k subsets such that the sum of the weights of edges joining vertices in different subsets is maximized. We focus on identifying effective classes of inequalities to tighten the semidefinite programming relaxation. We carry out an experimental study … Read more
The k-partition problem is an NP-hard combinatorial optimisation problem with many applications. Chopra and Rao introduced two integer programming formulations of this problem, one having both node and edge variables, and the other having only edge variables. We show that, if we take the polytopes associated with the `edge-only’ formulation, and project them into a … Read more