Numerical Investigation of Crouzeix’s Conjecture
Crouzeix’s conjecture states that for all polynomials p and matrices A, the inequality ||p(A)||
Homotopy Smoothing for Non-Smooth Problems with Lower Complexity than O(1/epsilon)
In this paper, we develop a novel {\bf ho}moto{\bf p}y {\bf s}moothing (HOPS) algorithm for solving a family of non-smooth problems that is composed of a non-smooth term with an explicit max-structure and a smooth term or a simple non-smooth term whose proximal mapping is easy to compute. The best known iteration complexity for solving … Read more
RSG: Beating Subgradient Method without Smoothness and Strong Convexity
In this paper, we study the efficiency of a {\bf R}estarted {\bf S}ub{\bf G}radient (RSG) method that periodically restarts the standard subgradient method (SG). We show that, when applied to a broad class of convex optimization problems, RSG method can find an $\epsilon$-optimal solution with a low complexity than SG method. In particular, we first … Read more
Generalized Symmetric ADMM for Separable Convex Optimization
The Alternating Direction Method of Multipliers (ADMM) has been proved to be effective for solving separable convex optimization subject to linear constraints. In this paper, we propose a Generalized Symmetric ADMM (GS-ADMM), which updates the Lagrange multiplier twice with suitable stepsizes, to solve the multi-block separable convex programming. This GS-ADMM partitions the data into two … Read more
A Primal-Dual Homotopy Algorithm for l_1-Minimization with l_inf-Constraints
In this paper we propose a primal-dual homotopy method for $\ell_1$-minimization problems with infinity norm constraints in the context of sparse reconstruction. The natural homotopy parameter is the value of the bound for the constraints and we show that there exists a piecewise linear solution path with finitely many break points for the primal problem … Read more
A Complete Characterization of Disjunctive Conic Cuts for Mixed Integer Second Order Cone Optimization
We study the convex hull of the intersection of a disjunctive set defined by parallel hyperplanes and the feasible set of a mixed integer second order cone optimization problem. We extend our prior work on disjunctive conic cuts, which has thus far been restricted to the case in which the intersection of the hyperplanes and … Read more
Lattice closures of polyhedra
Given $P\subset\R^n$, a mixed-integer set $P^I=P\cap (\Z^{t}\times\R^{n-t}$), and a $k$-tuple of $n$-dimensional integral vectors $(\pi_1, \ldots, \pi_k)$ where the last $n-t$ entries of each vector is zero, we consider the relaxation of $P^I$ obtained by taking the convex hull of points $x$ in $P$ for which $ \pi_1^Tx,\ldots,\pi^T_kx$ are integral. We then define the $k$-dimensional … Read more
An Inexact Spingarn’s Partial Inverse Method with Applications to Operator Splitting and Composite Optimization
We propose and study the iteration-complexity of an inexact version of the Spingarn’s partial inverse method. Its complexity analysis is performed by viewing it in the framework of the hybrid proximal extragradient (HPE) method, for which pointwise and ergodic iteration-complexity has been established recently by Monteiro and Svaiter. As applications, we propose and analyze the … Read more
A proximal-Newton method for unconstrained convex optimization in Hilbert spaces
We propose and study the iteration-complexity of a proximal-Newton method for finding approximate solutions of the problem of minimizing a twice continuously differentiable convex function on a (possibly infinite dimensional) Hilbert space. We prove global convergence rates for obtaining approximate solutions in terms of function/gradient values. Our main results follow from an iteration-complexity study of … Read more
An exact approach for the 0–1 Knapsack Problem with Setups
We consider the 0–1 Knapsack Problem with Setups. We propose an exact approach which handles the structure of the ILP formulation of the problem. It relies on partitioning the variables set into two levels and exploiting this partitioning. The proposed approach favorably compares to the algorithms in literature and to solver CPLEX 12.5 applied to … Read more