A unifying framework for the analysis of projection-free first-order methods under a sufficient slope condition

The analysis of projection-free first order methods is often complicated by the presence of different kinds of “good” and “bad” steps. In this article, we propose a unifying framework for projection-free methods, aiming to simplify the converge analysis by getting rid of such a distinction between steps. The main tool employed in our framework is … Read more

A derivative-free method for structured optimization problems

Structured optimization problems are ubiquitous in fields like data science and engineering. The goal in structured optimization is using a prescribed set of points, called atoms, to build up a solution that minimizes or maximizes a given function. In the present paper, we want to minimize a black-box function over the convex hull of a … Read more

Mining for diamonds – matrix generation algorithms for binary quadratically constrained quadratic problems

In this paper, we consider binary quadratically constrained quadratic problems and propose a new approach to generate stronger bounds than the ones obtained using the Semidefinite Programming relaxation. The new relaxation is based on the Boolean Quadric Polytope and is solved via a Dantzig-Wolfe Reformulation in matrix space. For block-decomposable problems, we extend the relaxation … Read more

Solving non-monotone equilibrium problems via a DIRECT-type approach

A global optimization approach for solving non-monotone equilibrium problems (EPs) is proposed. The class of (regularized) gap functions is used to reformulate any EP as a constrained global optimization program and some bounds on the Lipschitz constant of such functions are provided. The proposed global optimization approach is a combination of an improved version of … Read more

Zero Order Stochastic Weakly Convex Composite Optimization

In this paper we consider stochastic weakly convex composite problems, however without the existence of a stochastic subgradient oracle. We present a derivative free algorithm that uses a two point approximation for computing a gradient estimate of the smoothed function. We prove convergence at a similar rate as state of the art methods, however with … Read more

Active Set Complexity of the Away-step Frank-Wolfe Algorithm

In this paper, we study active set identification results for the away-step Frank-Wolfe algorithm in different settings. We first prove a local identification property that we apply, in combination with a convergence hypothesis, to get an active set identification result. We then prove, in the nonconvex case, a novel O(1/ √k) convergence rate result and … Read more

Improved Penalty Algorithm for Mixed Integer PDE Constrained Optimization (MIPDECO) Problems

Optimal control problems including partial differential equation (PDE) as well as integer constraints merge the combinatorial difficulties of integer programming and the challenges related to large-scale systems resulting from discretized PDEs. So far, the Branch-and-Bound framework has been the most common solution strategy for such problems. In order to provide an alternative solution approach, especially … Read more

Trust-region methods for the derivative-free optimization of nonsmooth black-box functions

In this paper we study the minimization of a nonsmooth black-box type function, without assuming any access to derivatives or generalized derivatives and without any knowledge about the analytical origin of the function nonsmoothness. Directional methods have been derived for such problems but to our knowledge no model-based method like a trust-region one has yet … Read more

Scanning integer points with lex-inequalities: A finite cutting plane algorithm for integer programming with linear objective

We consider the integer points in a unimodular cone K ordered by a lexicographic rule defined by a lattice basis. To each integer point x in K we associate a family of inequalities (lex-inequalities) that defines the convex hull of the integer points in K that are not lexicographically smaller than x. The family of … Read more

First-order methods for the impatient: support identification in finite time with convergent Frank-Wolfe variants

In this paper, we focus on the problem of minimizing a non-convex function over the unit simplex. We analyze two well-known and widely used variants of the Frank-Wolfe algorithm and first prove global convergence of the iterates to stationary points both when using exact and Armijo line search. Then we show that the algorithms identify … Read more