We introduce a polynomial time algorithm for optimizing the class of star-convex functions, under no Lipschitz or other smoothness assumptions whatsoever, and no restrictions except exponential boundedness on a region about the origin, and Lebesgue measurability. The algorithm’s performance is polynomial in the requested number of digits of accuracy and the dimension of the search domain. This ...

Aproximity theorem is astatement that, given an optimization problem and its relaxation, an optimal solution to the original problem exists in acertain neighborhood of asolution to the relaxation. Proximity theorems have been used successfully, for example, in designing efficient algorithms for discrete resource allocation problems. After reviewing the recent results for $\mathrm{L}$-convex and...

We address the problem of solving convex optimization problems with many convex constraints in a distributed setting. Our approach is based on an extension of the alternating direction method of multipliers (ADMM) that recently gained a lot of attention in the Big Data context. Although it has been invented decades ago, ADMM so far can be applied only to unconstrained problems and problems with...

We consider the convex optimization problem P : minx{f(x) : x ∈ K} where f is convex continuously differentiable, and K ⊂ R is a compact convex set with representation {x ∈ R : gj(x) ≥ 0, j = 1, . . . ,m} for some continuously differentiable functions (gj). We discuss the case where the gj ’s are not all concave (in contrast with convex programming where they all are). In particular, even if th...

Given a nite number of closed convex sets whose algebraic representation is known, we study the problem of optimizing a convex function over the closure of the convex hull of the union of these sets. We derive an algebraic characterization of the feasible region in a higher-dimensional space and propose a solution procedure akin to the interior-point approach for convex programming.

In this lecture we shall look at a fairly general setting of online convex optimization which, as we shall see, encompasses some of the online learning problems we have seen so far as special cases. A standard (offline) convex optimization problem involves a convex set Ω ⊆ R and a fixed convex cost function c : Ω 7→ R; the goal is to find a point x∗ ∈ Ω that minimizes c(x) over Ω. An online con...

In this paper, we first present a new important property for Bouligand tangent cone (contingent cone) of a star-shaped set. We then establish optimality conditions for Pareto minima and proper ideal efficiencies in nonsmooth vector optimization problems by means of Bouligand tangent cone of image set, where the objective is generalized cone convex set-valued map, in general real normed spaces.

Here we study non-convex composite optimization: first, a finite-sum of smooth but non-convex functions, and second, a general function that admits a simple proximal mapping. Most research on stochastic methods for composite optimization assumes convexity or strong convexity of each function. In this paper, we extend this problem into the non-convex setting using variance reduction techniques, ...