A New Dual-Based Cutting Plane Algorithm for Nonlinear Adjustable Robust Optimization

This paper explores a class of nonlinear Adjustable Robust Optimization (ARO) problems, containing here-and-now and wait-and-see variables, with uncertainty in the objective function and constraints. By applying Fenchel’s duality on the wait-and-see variables, we obtain an equivalent dual reformulation, which is a nonlinear static robust optimization problem. Using the dual formulation, we provide conditions under … Read more

Adaptive robust optimization with objective uncertainty

In this work, we study optimization problems where some cost parameters are not known at decision time and the decision flow is modeled as a two-stage process within a robust optimization setting. We address general problems in which all constraints (including those linking the first and the second stages) are defined by convex functions and … Read more

Generalized conditional subgradient and generalized mirror descent: duality, convergence, and symmetry

We provide new insight into a generalized conditional subgradient algorithm and a generalized mirror descent algorithm for the convex minimization problem \[\min_x \; \{f(Ax) + h(x)\}.\] As Bach showed in [SIAM J. Optim., 25 (2015), pp. 115–129], applying either of these two algorithms to this problem is equivalent to applying the other one to its … Read more

The Proximal Alternating Minimization Algorithm for two-block separable convex optimization problems with linear constraints

The Alternating Minimization Algorithm (AMA) has been proposed by Tseng to solve convex programming problems with two-block separable linear constraints and objectives, whereby (at least) one of the components of the latter is assumed to be strongly convex. The fact that one of the subproblems to be solved within the iteration process of AMA does … Read more

ADMM for monotone operators: convergence analysis and rates

We propose in this paper a unifying scheme for several algorithms from the literature dedicated to the solving of monotone inclusion problems involving compositions with linear continuous operators in infinite dimensional Hilbert spaces. We show that a number of primal-dual algorithms for monotone inclusions and also the classical ADMM numerical scheme for convex optimization problems, … 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 universal and structured way to derive dual optimization problem formulations

The dual problem of a convex optimization problem can be obtained in a relatively simple and structural way by using a well-known result in convex analysis, namely Fenchel’s duality theorem. This alternative way of forming a strong dual problem is the subject in this paper. We recall some standard results from convex analysis and then … Read more

An inertial alternating direction method of multipliers

In the context of convex optimization problems in Hilbert spaces, we induce inertial effects into the classical ADMM numerical scheme and obtain in this way so-called inertial ADMM algorithms, the convergence properties of which we investigate into detail. To this aim we make use of the inertial version of the Douglas-Rachford splitting method for monotone … Read more

A Douglas-Rachford type primal-dual method for solving inclusions with mixtures of composite and parallel-sum type monotone operators

In this paper we propose two different primal-dual splitting algorithms for solving inclusions involving mixtures of composite and parallel-sum type monotone operators which rely on an inexact Douglas-Rachford splitting method, however applied in different underlying Hilbert spaces. Most importantly, the algorithms allow to process the bounded linear operators and the set-valued operators occurring in the … Read more

Convergence analysis for a primal-dual monotone + skew splitting algorithm with applications to total variation minimization

In this paper we investigate the convergence behavior of a primal-dual splitting method for solving monotone inclusions involving mixtures of composite, Lipschitzian and parallel sum type operators proposed by Combettes and Pesquet in [7]. Firstly, in the particular case of convex minimization problems, we derive convergence rates for the sequence of objective function values by … Read more