Claudia Sagastizábal

A Regularized Smoothing Method for Fully Parameterized Convex Problems with Applications to Convex and Nonconvex Two-Stage Stochastic Programming

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

We present an approach to regularize and approximate solution mappings of parametric convex optimization problems that combines interior penalty (log-barrier) solutions with Tikhonov regularization. Because the regularized mappings are single-valued and smooth under reasonable conditions, they can be used to build a computationally practical smoothing for the associated optimal value function. The value function in … Read more

Quantifying the value of flexibility: demand response versus storage

Categories Energy, Multi-Criteria Optimization

Intermittent sources of energy represent a challenge for electrical networks, particularly regarding demand satisfaction at peak times. Energy management tools such as load shaving or storage systems can be used to mitigate abrupt variations in the network.The value of different mechanisms to move energy through time is determined by a multi-objective programming approach, that aims … Read more

Stochastic Hydro-thermal Unit Commitment via Multi-level Scenario Trees and Bundle Regularization

Categories Nonlinear Optimization, Stochastic Programming

For an electric power mix subject to uncertainty, the stochastic unit-commitment problem finds short-term optimal generation schedules that satisfy several system-wide constraints. In regulated electricity markets, this very practical and important problem is used by the system operator to decide when each unit is to be started or stopped, and to define how to generate … Read more

Probabilistic optimization via approximate p-efficient points and bundle methods

Categories Convex and Nonsmooth Optimization, Stochastic Programming

For problems when decisions are taken prior to observing the realization of underlying random events, probabilistic constraints are an important modelling tool if reliability is a concern. A key concept to numerically dealing with probabilistic constraints is that of p-efficient points. By adopting a dual point of view, we develop a solution framework that includes … Read more

Bundle methods in the XXIst century: A bird’s-eye view

Categories Convex Optimization Tags , ,

Bundle methods are often the algorithms of choice for nonsmooth convex optimization, especially if accuracy in the solution and reliability are a concern. We review several algorithms based on the bundle methodology that have been developed recently and that, unlike their forerunner variants, have the ability to provide exact solutions even if most of the … Read more

An approximation scheme for a class of risk-averse stochastic equilibrium problems

Categories Complementarity and Variational Inequalities, Stochastic Programming Tags , , , , ,

We consider two models for stochastic equilibrium: one based on the variational equilibrium of a generalized Nash game, and the other on the mixed complementarity formulation. Each agent in the market solves a one-stage risk-averse optimization problem with both here-and-now (investment) variables and (production) wait-and-see variables. A shared constraint couples almost surely the wait-and-see decisions … Read more

Proximal bundle methods in depth: a unified analysis for inexact oracles

Categories Convex and Nonsmooth Optimization Tags , ,

The last few years have seen the advent ofa new generation of bundle methods, capable to handle inexact oracles, polluted by “noise”. Proving convergence of a bundle method is never simple and coping with inexact oracles substantially increases the technicalities. Besides, several variants exist to deal with noise, each one needing an ad hoc proof … Read more

Constrained Bundle Methods for Upper Inexact Oracles with Application to Joint Chance Constrained Energy Problems

Categories Convex and Nonsmooth Optimization, Stochastic Programming Tags , ,

Joint chance constrained problems give rise to many algorithmic challenges. Even in the convex case, i.e., when an appropriate transformation of the probabilistic constraint is a convex function, its cutting-plane linearization is just an approximation, produced by an oracle providing subgradient and function values that can only be evaluated inexactly. As a result, the cutting-plane … Read more

Level Bundle Methods for oracles with on-demand accuracy

Categories Convex and Nonsmooth Optimization, Stochastic Programming Tags , , ,

For nonsmooth convex optimization, we consider level bundle methods built using an oracle that computes values for the objective function and a subgradient at any given feasible point. For the problems of interest, the exact oracle information is computable, but difficult to obtain. In order to save computational effort the oracle can provide estimations with … Read more

A Class of Dantzig-Wolfe Type Decomposition Methods for Variational Inequality Problems

Categories Complementarity and Variational Inequalities, Convex and Nonsmooth Optimization Tags ,

We consider a class of decomposition methods for variational inequalities, which is related to the classical Dantzig–Wolfe decomposition of linear programs. Our approach is rather general, in that it can be used with set-valued or nonmonotone operators, as well as various kinds of approximations in the subproblems of the functions and derivatives in the single-valued … Read more