Variational Analysis and Optimization of Sweeping Processes with Controlled Moving Sets

This paper briefly overviews some recent and very fresh results on a rather new class of dynamic optimization problems governed by the so-called sweeping (Moreau) processes with controlled moving sets. Uncontrolled sweeping processes have been known in dynamical systems and applications starting from 1970s while control problems for them have drawn attention of mathematicians, applied … Read more

A global hybrid derivative-free method for large-scale systems of nonlinear equations

This work concerns the numerical solution of large-scale systems of nonlinear equations, when derivatives are not available for use, but assuming that all functions defining the problem are continuously differentiable. A hybrid approach is taken, based on a derivative-free iterative method, organized in three phases. The first phase is defined by derivative-free versions of a … Read more

An Envelope for Davis-Yin Splitting and Strict Saddle Point Avoidance

It is known that operator splitting methods based on Forward Backward Splitting (FBS), Douglas-Rachford Splitting (DRS), and Davis-Yin Splitting (DYS) decompose a difficult optimization problems into simpler subproblem under proper convexity and smoothness assumptions. In this paper, we identify an envelope (an objective function) whose gradient descent iteration under a variable metric coincides with DYS … Read more

On proximal point-type algorithms for weakly convex functions and their connection to the backward Euler method

In this article we study the connection between proximal point methods for nonconvex optimization and the backward Euler method from numerical Ordinary Differential Equations (ODEs). We establish conditions for which these methods are equivalent. In the case of weakly convex functions, for small enough parameters, the implicit steps can be solved using a strongly convex … Read more

Refining the partition for multifold conic optimization problems

In this paper we give a unified treatment to different definitions of complementarity partition for a primal-dual pair of linear conic optimization problem. Citation Submitted ArXiv 1804.00386 Article Download View Refining the partition for multifold conic optimization problems

On a Frank-Wolfe Type Theorem in Cubic Optimization

A classical result due to Frank and Wolfe (1956) says that a quadratic function $f$ attains its supremum on a nonempty polyhedron $M$ if $f$ is bounded from above on $M$. In this note, we present a stringent proof of the extension of this result to cubic optimization (known from Andronov, Belousov and Shironin (1982)). … Read more

Acyclic Mechanism Design for Freight Consolidation

Freight consolidation is a logistics practice that improves the cost-effectiveness and efficiency of transportation operations, and also reduces energy consumption and carbon footprint. A “fair” shipping cost sharing scheme is indispensable to help establish and sustain the cooperation of a group of suppliers in freight consolidation. In this paper, we design a truthful acyclic mechanism … Read more