New Multirectional Mean Value Inequality
We establish new and stronger inequality of Clarke-Ledyaev type by direct construction. CitationPreprint, 2017, Sofia University ArticleDownload View PDF
Convex and Nonsmooth Optimization
Geometric descent method for convex composite minimization
In this paper, we extend the geometric descent method recently proposed by Bubeck, Lee and Singh to tackle nonsmooth and strongly convex composite problems. We prove that our proposed algorithm, dubbed geometric proximal gradient method (GeoPG), converges with a linear rate $(1-1/\sqrt{\kappa})$ and thus achieves the optimal rate among first-order methods, where $\kappa$ is the … Read more
An extension of Chubanov’s algorithm to symmetric cones
In this work we present an extension of Chubanov’s algorithm to the case of homogeneous feasibility problems over a symmetric cone K. As in Chubanov’s method for linear feasibility problems, the algorithm consists of a basic procedure and a step where the solutions are confined to the intersection of a half-space and K. Following an … Read more
The p-cones in dimension n>=3 are not homogeneous when p \neq 2
Using the T-algebra machinery we show that the only strictly convex homogeneous cones in R^n with n >= 3 are the 2-cones, also known as Lorentz cones or second order cones. In particular, this shows that the p-cones are not homogeneous when p is not 2, 1 < p <\infty and n >= 3, thus … Read more
A Limited-Memory Quasi-Newton Algorithm for Bound-Constrained Nonsmooth Optimization
We consider the problem of minimizing a continuous function that may be nonsmooth and nonconvex, subject to bound constraints. We propose an algorithm that uses the L-BFGS quasi-Newton approximation of the problem’s curvature together with a variant of the weak Wolfe line search. The key ingredient of the method is an active-set selection strategy that … Read more
Efficiency of minimizing compositions of convex functions and smooth maps
We consider the problem of minimizing a sum of a convex function and a composition of a convex function with a smooth map. Important examples include exact penalty formulations of nonlinear programs and nonlinear least squares problems with side constraints. The basic algorithm we rely on is the well-known prox-linear method, which in each iteration … Read more
Fixing and extending some recent results on the ADMM algorithm
We first point out several flaws in the recent paper {\it [R. Shefi, M. Teboulle: Rate of convergence analysis of decomposition methods based on the proximal method of multipliers for convex minimization, SIAM J. Optim. 24, 269–297, 2014]} that proposes two ADMM-type algorithms for solving convex optimization problems involving compositions with linear operators and show … Read more
A Primal-dual Three-operator Splitting Scheme
In this paper, we propose a new primal-dual algorithm for minimizing f(x)+g(x)+h(Ax), where f, g, and h are convex functions, f is differentiable with a Lipschitz continuous gradient, and A is a bounded linear operator. It has some famous primal-dual algorithms for minimizing the sum of two functions as special cases. For example, it reduces … Read more
An Algorithm for Nonsmooth Optimization by Successive Piecewise Linearization
We present an optimization method for Lipschitz continuous, piecewise smooth (PS) objective functions based on successive piecewise linearization. Since, in many realistic cases, nondifferentiabilities are caused by the occurrence of abs(), max(), and min(), we concentrate on these nonsmooth elemental functions. The method’s idea is to locate an optimum of a PS objective function by … Read more
Expander Graph and Communication-Efficient Decentralized Optimization
In this paper, we discuss how to design the graph topology to reduce the communication complexity of certain algorithms for decentralized optimization. Our goal is to minimize the total communication needed to achieve a prescribed accuracy. We discover that the so-called expander graphs are near-optimal choices. We propose three approaches to construct expander graphs for … Read more