We show that the evolution of a dynamical system driven by controls obtained by the solution of an embedded optimization problem (as done in MPC) can be cast as a differential variational inequality (DVI). The DVI abstraction reveals that standard sampled-data MPC implementations (in which the control law is computed using states that are sampled … Read more
The sparse bounded degree sum-of-squares (sparse-BSOS) hierarchy of Weisser, Lasserre and Toh [arXiv:1607.01151,2016] constructs a sequence of lower bounds for a sparse polynomial optimization problem. Under some assumptions, it is proven by the authors that the sequence converges to the optimal value. In this paper, we modify the hierarchy to deal with problems containing equality … Read more
We consider the problem of minimizing a continuous function f over a compact set K. We compare the hierarchy of upper bounds proposed by Lasserre in [SIAM J. Optim. 21(3) (2011), pp. 864-885] to bounds that may be obtained from simulated annealing. We show that, when f is a polynomial and K a convex body, … Read more
We prove an extension of Yuan’s Lemma to more than two matrices, as long as the set of matrices has rank at most 2. This is used to generalize the main result of [A. Baccari and A. Trad. On the classical necessary second-order optimality conditions in the presence of equality and inequality constraints. SIAM J. … Read more
In this paper, an augmented Lagrangian proximal alternating (ALPA) method is proposed for two class of large-scale sparse discrete constrained optimization problems in which a sequence of augmented Lagrangian subproblems are solved by utilizing proximal alternating linearized minimization framework and sparse projection techniques. Under the Mangasarian-Fromovitz and the basic constraint qualification, we show that any … Read more
This paper addresses the question of whether it can be beneficial for an optimization algorithm to follow directions of negative curvature. Although some prior work has established convergence results for algorithms that integrate both descent and negative curvature directions, there has not yet been numerical evidence showing that such methods offer significant performance improvements. In … Read more
This paper studies a class of problems consisting of minimizing a continuously differentiable function penalized with the so-called $\ell_0$-norm over a symmetric set. These problems are hard to solve, yet prominent in many fields and applications. We first study the proximal mapping with respect to the $\ell_0$-norm over symmetric sets, and provide an efficient method … Read more
Trust-region algorithms have been proved to globally converge with probability one when the accuracy of the trust-region models is imposed with a certain probability conditioning on the iteration history. In this paper, we study their complexity, providing global rates and worst case complexity bounds on the number of iterations (with overwhelmingly high probability), for both … Read more
Direct search is a methodology for derivative-free optimization whose iterations are characterized by evaluating the objective function using a set of polling directions. In deterministic direct search applied to smooth objectives, these directions must somehow conform to the geometry of the feasible region and typically consist of positive generators of approximate tangent cones (which then … Read more
The optimization of multimodal functions is a challenging task, in particular when derivatives are not available for use. Recently, in a directional direct search framework, a clever multistart strategy was proposed for global derivative-free optimization of single objective functions. The goal of the current work is to generalize this approach to the computation of global … Read more