# Claudia SagastizabalNational Institute for Research in Computer Science and Control | INRIA · IMPA, visiting researcher

Claudia Sagastizabal

Dr, HdR

## About

162

Publications

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4,214

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Introduction

## Publications

Publications (162)

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 c...

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 smooth...

We develop new variants of Benders decomposition methods for variational inequality problems. The construction is done by applying the general class of Dantzig–Wolfe decomposition of Luna et al. (Math Program 143(1–2):177–209, 2014) to an appropriately defined dual of the given variational inequality, and then passing back to the primal space. As c...

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 d...

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 intermittency. In this work, the value of different mechanisms to move energy through time is examined through a multi-objecti...

When minimizing a nonsmooth convex function bundle methods are well known by their robustness and reliability. While such features are related to global convergence properties of the algorithm, speed of convergence is a different matter, as fast rates cannot be expected if the objective function lacks smoothness.

In many mathematical optimization applications, dual variables are an important output of the solving process, due to their role as price signals. When dual solutions are not unique, different solvers or different computers, even different runs in the same computer if the problem is stochastic, often end up with different optimal multipliers. From...

In Variational Analysis, VU-theory provides a set of tools that is helpful for understanding and exploiting the structure of nonsmooth functions. The theory takes advantage of the fact that at any point, the space can be separated into two orthogonal subspaces: one that describes the direction of nonsmoothness of the function, and the other on whic...

The $\mathcal{VU}$-algorithm is a superlinearly convergent method for minimizing nonsmooth, convex functions. At each iteration, the algorithm works with a certain $\mathcal{V}$-space and its orthogonal $\U$-space, such that the nonsmoothness of the objective function is concentrated on its projection onto the $\mathcal{V}$-space, and on the $\math...

We review the concept of \(\mathcal {V}\mathcal {U}\)-decomposition of nonsmooth convex functions, which is closely related to the notion of partly smooth functions. As \(\mathcal {V}\mathcal {U}\)-decomposition depends on the subdifferential at the given point, the associated objects lack suitable continuity properties (because the subdifferential...

To describe the joint dynamics of prices of crude oil and refined products we extend two-factor models to a multidimensional setting. The new model captures directly the general correlation structure between the different commodities in the form of certain covariance matrix. Since the associated state-space formulation makes use of such correlation...

In Variational Analysis, VU-theory provides a set of tools that is helpful for understanding and exploiting the structure of nonsmooth functions. The theory takes advantage of the fact that at any point, the space can be separated into two orthogonal subspaces, one that describes the direction of nonsmoothness of the function, and the other on whic...

The VU-algorithm is a superlinearly convergent method for minimizing nonsmooth convex functions. At each iteration, the algorithm separates R n into the V-space and the orthogonal U-space, such that the nonsmoothness of the objective function is concentrated on its projection onto the V-space, and on the U-space the projection is smooth. This struc...

http://paperity.org/p/77751146/editorial-for-the-special-issue-optimization-in-energy

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 framew...

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 single-stage risk-averse optimization problem with both here-and-now (investment) variables and (production) wait-and-see variables. A share...

This project deals with theoretical and numerical studies aiming at the improvement of unit-commitment problems subject to uncertainty. Unit-commitment refers to the operation of a physical system - the power mix - for which it is becoming more and more important to incorporate stochastic effects, due to the increased use of renewable energy source...

For a class of nonconvex nonsmooth functions, we consider the problem of computing an approximate critical point, in the case when only inexact information about the function and subgradient values is available. We assume that the errors in function and subgradient evaluations are merely bounded, and in principle need not vanish in the limit. We ex...

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...

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 estim...

Liquefied Natural Gas contracts offer cancelation options that make their pricing difficult, especially if many gas storages need to be taken into account. We develop a valuation mechanism from the buyer’s perspective, a large gas company whose main interest in these contracts is to provide to clients a reliable supply of gas. The approach combines...

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 inexa...

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 certain types of set-valued or nonmonotone operators, as well as with various kinds of approximations in the subproblems of the funct...

Electricity and natural gas transmission and distribution networks are subject to regulation in price, service quality, and emission limits. The interaction of competing agents in an energy market subject to various regulatory interventions is usually modeled through equilibrium problems that ensure profit maximization for all the agents. These typ...

The last few years have seen the advent of a 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...

We discuss the energy generation expansion planning with environmental constraints, formulated as a nonsmooth convex constrained optimization problem. To solve such problems, methods suitable for constrained nonsmooth optimization need to be employed. We describe a recently developed approach, which applies the usual unconstrained bundle techniques...

We consider an interstage dependent stochastic process whose components follow an autoregressive model with time varying order. At a given time, we give some recursive formulæ linking future values of the process with past values and noises. We then consider multistage stochastic linear programs with uncertain sets depending affinely on such proces...

We consider minimization of nonsmooth functions which can be represented as the composition of a positively homogeneous convex function and a smooth mapping. This is a suciently rich class that includes max-functions, largest eigenvalue functions, and norm-1 regularized functions. The bundle method uses an oracle that is able to compute separately...

Modern electricity systems provide a plethora of challenging issues in optimization. The increasing penetration of low carbon renewable sources of energy introduces uncertainty in problems traditionally modeled in a deterministic setting. The liberalization of the electricity sector brought the need of designing sound markets, ensuring capacity inv...

We consider risk-averse formulations of stochastic linear programs having a structure that is common in real-life applications. Specifically, the optimization problem corresponds to controlling over a certain horizon a system whose dynamics is given by a transition equation depending affinely on an interstage dependent stochastic process. We put in...

We consider the optimal management of a hydro-thermal power system in the mid and long terms. From the optimization point of view, this amounts to a large-scale multistage stochastic linear program, often solved by combining sampling with decomposition algorithms, like stochastic dual dynamic programming. Such methodologies, however, may entail pro...

Stochastic programming problems arise in many practical situations. In general, the deterministic equivalents of these problems can be very large and may not be solvable directly by general-purpose optimization approaches. For the particular case of two-stage stochastic programs, we consider decomposition approaches akin to a regularized L-shaped m...

We consider the problem of optimally determining an investment portfolio for an energy company owning a network of gas pipelines,
and in charge of purchasing, selling and distributing gas. We propose a two stage stochastic investment model which hedges
risk by means of Conditional Value at Risk constraints. The model, solved by a decomposition meth...

1. Abstract For the sake of precision, mid-term operation planning of hydro-thermal power systems needs a large number of synthetic sequences to represent accurately stochastic streamflows. However, if the number of synthetic sequences is too big, the optimization planning problem may be too difficult, due to com- putational time. This work employs...

An important field of application of non-smooth optimization refers to decomposition of large-scale or complex problems by
Lagrangian duality. In this setting, the dual problem consists in maximizing a concave non-smooth function that is defined
as the sum of sub-functions. The evaluation of each sub-function requires solving a specific optimizatio...

Expanding an electrical transmission network requires heavy investments that need to be carefully planned, often at a regional
or national level. We study relevant theoretical and practical aspects of transmission expansion planning, set as a bilinear
programming problem with mixed 0–1 variables. We show that the problem is NP-hard and that, unlike...

Proximal bundle methods have been shown to be highly successful optimization methods for unconstrained convex problems with discontinuous first derivatives. This naturally leads to the question of whether proximal variants of bundle methods can be extended to a nonconvex setting. This work proposes an approach based on generating cutting-planes mod...

Lagrangian relaxation is a popular technique to solve difficult optimization problems. However, the applicability of this
technique depends on having a relatively low number of hard constraints to dualize. When there are many hard constraints,
it may be preferable to relax them dynamically, according to some rule depending on which multipliers are...

The proximal point mapping is the basis of many optimization techniques for convex functions. By means of variational analysis,
the concept of proximal mapping was recently extended to nonconvex functions that are prox-regular and prox-bounded. In such
a setting, the proximal point mapping is locally Lipschitz continuous and its set of fixed points...

For solving nonsmooth convex constrained optimization problems, we propose an algorithm which combines the ideas of the proximal bundle methods with the filter strategy for evaluating candidate points. The resulting algorithm inherits some attractive features from both approaches. On the one hand, it allows effective control of the size of quadrati...

We consider the problem of minimizing nonsmooth convex functions, dened piecewise by a nite number of functions each of which is either convex quadratic or twice continuously dieren tiable with positive denite Hessian on the set of interest. This is a particular case of functions with primal-dual gradient structure, a notion closely related to the...

Keywords
Synonyms
Examples of the Problem
Continuity and Optimality Conditions
Algorithms of Minimization
Nonlinear Programming
Nonsmooth Optimization
Other Methods of Resolution
See also
References

Thermal unit commitment problems are encountered often when solving short term scheduling problems. In this short term context, in order to model accurately the different status for the thermal unit along the planning horizon, it is important to include time-dependent startup costs, up/down time, and ramping constraints. We consider two solution st...

Lagrangian relaxation (LR) is one of the most widely used techniques to solve the short-term hydrothermal scheduling problem (HTS) for large power systems. The classical form of LR relaxes coupling constraints such as demand and reserve requirements for the whole mix. However, since with this form of relaxation the number of dual variables increase...

Hydropower generation is a function of the discharge of the generating units and the difference between the forebay and tailrace levels of the reservoir, and is subject to penstock head losses and to the generation unit efficiency factor. This factor is in turn a function of the turbined flow and the head in the reservoir, and is usually obtained f...

Generation expansion planning problems should take into account not only satisfaction of future demand but also environmental constraints, such as government imposed limits on pollution produced by coal-based power plants. For a vertically integrated system of power plants, we model those features as a two-stage stochastic programming problem with...

For a maximal monotone operator T on a Hilbert space H and a closed subspace A of H, we consider the problem of finding (x, y∈T(x)) satisfying x∈A and y∈A . An equivalent formulation of this problem makes use of the partial inverse operator of Spingarn. The resulting generalized equation can be solved by using the proximal point algorithm. We consi...

To minimize a convex function f, we state a penalty-type bundle algorithm, where the penalty uses a variable metric. This metric is updated according to quasi-Newton formulae based on Moreau-Yosida approximations of f. In particular, we introduce a reversal quasi-Newton formula, specially suited for our purpose. We consider several variants in the...

The major focus of this work is to compare several methods for computing the proximal point of a nonconvex function via numerical testing. To do this, we introduce two techniques for randomly generating challenging nonconvex test functions, as well as two very specific test functions which should be of future interest to Nonconvex Optimization Benc...

For a maximal monotone operator T on a Hilbert space H and a closed subspace A of H, we consider the problem of finding (x, y ∈ T (x)) satisfying x ∈ A and y ∈ A⊥. An equivalent formulation of this problem makes use of the partial inverse operator of Spingarn. The resulting generalized equation can be solved by using the proximal point algorithm. W...