Welington de Oliveira

A Computational Toolbox for Linear Optimization with Joint Affine Chance Constraints

Categories Constrained Nonlinear Optimization, Linear Programming, Optimization Software Benchmark Tags , , , ,

We present a Julia computational toolbox for linear optimization problems with joint affine chance constraints under elliptically symmetric uncertainty. The toolbox combines a spherical–radial oracle for estimating the joint probability and its gradient with three structured optimization methods: Proximal, Feasible, and Penalty. The oracle supports several elliptically symmetric distributions and is integrated with these methods … Read more

A Proximal Approach for Nonsmooth Composite-Constrained Optimization

Categories Constrained Nonlinear Optimization, Nonlinear Optimization, Nonsmooth Optimization, Stochastic Programming Tags , , ,

We propose a proximal-type algorithm for nonsmooth and nonconvex optimization problems with composite constraints. The constraint is defined by the composition of a locally upper-\(C^2\) outer function with a locally Lipschitz continuous inner mapping. The method is based on an improvement function that balances objective decrease and constraint satisfaction, and on a surrogate model obtained … Read more

Lipschitz Gradient Guarantees for Probability Functions and a New Algorithm for Probability Maximization

Categories Stochastic Programming Tags , , ,

This work studies probability functions that appear in stochastic programming models. Although their differentiability has been widely investigated, the Lipschitz continuity of their gradients, crucial for the design and analysis of modern optimization algorithms, has received little attention. We develop a general framework that ensures differentiability and gradient Lipschitz continuity under practical conditions. Our framework … Read more

A progressive decoupling algorithm for minimizing the difference of convex and weakly convex functions over a linear subspace

Categories Nonlinear Optimization, Nonsmooth Optimization Tags , ,

Commonly, decomposition and splitting techniques for optimization problems strongly depend on convexity. Implementable splitting methods for nonconvex and nonsmooth optimization problems are scarce and often lack convergence guarantees. Among the few exceptions is the Progressive Decoupling Algorithm (PDA), which has local convergence should convexity be elicitable. In this work, we furnish PDA with a descent … Read more

Minimizing the difference of convex and weakly convex functions via bundle method

Categories Constrained Nonlinear Optimization, Nonlinear Optimization Tags , , ,

We consider optimization problems with objective and constraint being the difference of convex and weakly convex functions. This framework covers a vast family of nonsmooth and nonconvex optimization problems, particularly those involving certain classes of composite and nonconvex value functions. We investigate several stationary conditions and extend the proximal bundle algorithm of [van Ackooij et … Read more

Revisiting Augmented Lagrangian Duals

Categories Nonsmooth Optimization Tags , ,

For nonconvex optimization problems, possibly having mixed-integer variables, a convergent primal-dual solution algorithm is proposed. The approach applies a proximal bundle method to certain augmented Lagrangian dual that arises in the context of the so-called generalized augmented Lagrangians. We recast these Lagrangians into the framework of a classical Lagrangian, by means of a special reformulation … Read more

A bundle method for nonsmooth DC programming with application to chance-constrained problems

Categories Constrained Nonlinear Optimization, Nonsmooth Optimization Tags

This work considers nonsmooth and nonconvex optimization problems whose objective and constraint functions are defined by difference-of-convex (DC) functions. We consider an infeasible bundle method based on the so-called improvement functions to compute critical points for problems of this class. Our algorithm neither employs penalization techniques nor solves subproblems with linearized constraints. The approach, which … Read more

On level regularization with normal solutions in decomposition methods for multistage stochastic programming problems

Categories Nonsmooth Optimization, Stochastic Programming Tags , , ,

We consider well-known decomposition techniques for multistage stochastic programming and a new scheme based on normal solutions for stabilizing iterates during the solution process. The given algorithms combine ideas from finite perturbation of convex programs and level bundle methods to regularize the so-called forward step of these decomposition methods. Numerical experiments on a hydrothermal scheduling … Read more

An Adaptive Partition-based Level Decomposition for Solving Two-stage Stochastic Programs with Fixed Recourse

Categories Stochastic Programming

We present a computational study of several strategies to solve two-stage stochastic linear programs by integrating the adaptive partition-based approach with level decomposition. A partition-based formulation is a relaxation of the original stochastic program, obtained by aggregating variables and constraints according to a scenario partition. Partition refinements are guided by the optimal second-stage dual vectors … Read more

A Bundle Method for Exploiting Additive Structure in Difficult Optimization Problems

Categories Convex Optimization, Nonsmooth Optimization

This paper describes a bundle method for (approximately) minimizing complicated nonsmooth convex functions with additive structure, with the primary goal of computing bounds on the solution values of difficult optimization problems such as stochastic integer programs. The method combines features that have appeared in previously proposed bundle methods, but not in the particular configuration we … Read more