The authors’ paper in Ref. [5], was the first one to provide detailed optimality conditions for pessimistic bilevel optimization. The results there were based on the concept of the two-level optimal value function introduced and analyzed in Ref. [4], for the case of optimistic bilevel programs. One of the basic assumptions in both of these … Read more
In this paper, we propose and study the iteration complexity of an inexact Douglas-Rachford splitting (DRS) method and a Douglas-Rachford-Tseng’s forward-backward (F-B) splitting method for solving two-operator and four-operator monotone inclusions, respectively. The former method (although based on a slightly different mechanism of iteration) is motivated by the recent work of J. Eckstein and W. … Read more
Rencently, the generalized aternating direction method of multipliers (GADMM) proposed by Eckstein and Bertsekas has received intensive attention from a broad spectrum of areas. In this paper, we consider the convergence rate of GADMM when applying to the convex optimization problems that the subdifferentials of the underlying functions are piecewise linear multifunctions, including LASSO, a … Read more
We provide a framework for obtaining error bounds for linear conic problems without assuming constraint qualifications or regularity conditions. The key aspects of our approach are the notions of amenable cones and facial residual functions. For amenable cones, it is shown that error bounds can be expressed as a composition of facial residual functions. The … Read more
It has long been known that the gradient (steepest descent) method may fail on nonsmooth problems, but the examples that have appeared in the literature are either devised specifically to defeat a gradient or subgradient method with an exact line search or are unstable with respect to perturbation of the initial point. We give an … Read more
In this paper, we consider a class of finite-sum convex optimization problems defined over a distributed multiagent network with $m$ agents connected to a central server. In particular, the objective function consists of the average of $m$ ($\ge 1$) smooth components associated with each network agent together with a strongly convex term. Our major contribution … Read more
We address the problem of prescribing an optimal decision in a framework where its cost depends on uncertain problem parameters $Y$ that need to be learned from data. Earlier work by Bertsimas and Kallus (2014) transforms classical machine learning methods that merely predict $Y$ from supervised training data $[(x_1, y_1), \dots, (x_n, y_n)]$ into prescriptive … Read more
We study the existence and uniqueness of equilibria for perfectly competitive markets in capacitated transport networks. The model under consideration is rather general so that it captures basic aspects of related models in, e.g., gas or electricity networks. We formulate the market equilibrium model as a mixed complementarity problem and show the equivalence to a … Read more
The direct extension of the classic alternating direction method with multipliers (ADMMe) to the multi-block separable convex optimization problem is not necessarily convergent, though it often performs very well in practice. In order to preserve the numerical advantages of ADMMe and obtain convergence, many modified ADMM were proposed by correcting the output of ADMMe or … Read more
An abstract convergence theorem for a class of generalized descent methods that explicitly models relative errors is proved. The convergence theorem generalizes and unifies several recent abstract convergence theorems. It is applicable to possibly non-smooth and non-convex lower semi-continuous functions that satisfy the Kurdyka–Lojasiewicz (KL) inequality, which comprises a huge class of problems. Most of … Read more