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A survey of adjustable robust optimization

EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2019年第3期277卷 799-813页

作者： Yanikoglu, Ihsan Gorissen, Bram L. den Hertog, Dick Ozyegin Univ Dept Ind Engn Istanbul Turkey Free Univ Amsterdam Dept Math NL-1081 HV Amsterdam Netherlands Tilburg Univ Dept Econometr & Operat Res NL-5000 LE Tilburg Netherlands

Static robust optimization (RO) is a methodology to solve mathematical optimization problems with uncertain data. The objective of static RO is to find solutions that are immune to all perturbations of the data in a so-called uncertainty set. RO is popular because it is a computationally tractable methodology and has a wide range of applications in practice. Adjustable robust optimization (ARO), on the other hand, is a branch of RO where some of the decision variables can be adjusted after some portion of the uncertain data reveals itself. ARO generally yields a better objective function value than that in static robust optimization because it gives rise to more flexible adjustable (or wait-and-see) decisions. Additionally, ARO also has many real life applications and is a computationally tractable methodology for many parameterized adjustable decision variables and uncertainty sets. This paper surveys the state-of-the-art literature on applications and theoretical/methodological aspects of ARO. Moreover, it provides a tutorial and a road map to guide researchers and practitioners on how to apply ARO methods, as well as, the advantages and limitations of the associated methods. (C) 2018 Elsevier B.V. All rights reserved.

关键词： semi-infinite programming Robust optimization Adjustable robust optimization Multistage decision making

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Stability of the primal-dual partition in linear semi-infinite programming

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OPTIMIZATION 2012年第12期61卷 1449-1465页

作者： Ochoa, Pablo D. Vera de Serio, Virginia N. Univ Nacl Cuyo Fac Ciencias Econ Inst Ciencias Basicas RA-5500 Mendoza Argentina

We analyse the primal-dual states in linear semi-infinite programming (LSIP), where we consider the primal problem and the so called Haar's dual problem. Any linear programming problem and its dual can be classified as bounded, unbounded or inconsistent, giving rise to nine possible primal-dual states, which are reduced to six by the weak duality property. Recently, Goberna and Todorov have studied this partition and its stability in continuous LSIP in a series of papers [M. A. Goberna and M. I. Todorov, Primal, dual and primal-dual partitions in continuous linear semi-infinite programming, Optimization 56 (2007), pp. 617-628;M. A. Goberna and M. I. Todorov, Generic primal-dual solvability in continuous linear semi-infinite programming, Optimization 57 (2008), pp. 239-248]. In this article we consider the general case, with no continuity assumptions, discussing the maintenance of the primal-dual state of the problem by allowing small perturbations of the data. We characterize the stability of all of the six possible primal-dual states through necessary and sufficient conditions which depend on the data, and can be easily checked, showing some differences with the continuous case. These conditions involve the strong Slater constraint qualification, and some distinguished convex sets associated to the data.

关键词： duality primal-dual partition stability linear programming semi-infinite programming

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Decomposition algorithm for distributionally robust optimization using Wasserstein metric with an application to a class of regression models

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2019年第1期278卷 20-35页

作者： Luo, Fengqiao Mehrotra, Sanjay Northwestern Univ Dept Ind Engn & Management Sci Evanston IL 60202 USA

We study distributionally robust optimization (DRO) problems where the ambiguity set is defined using the Wasserstein metric and can account for a bounded support. We show that this class of DRO problems can be reformulated as decomposable semi-infinite programs. We use a cutting-surface method to solve the reformulated problem for the general nonlinear model, assuming that we have a separation oracle. As examples, we consider the problems arising from the machine learning models where variables couple with data in a bilinear form in the loss function. We present a branch-and-bound algorithm to solve the separation problem for this case using an iterative piece-wise linear approximation scheme. We use a distributionally robust generalization of the logistic regression model to test our algorithm. We also show that it is possible to approximate the logistic-loss function with significantly less linear pieces than those needed for a general loss function to achieve a given accuracy when generating a cutting surface. Numerical experiments on the distributionally robust logistic regression models show that the number of oracle calls are typically 20-50 to achieve 5-digit precision. The solution found by the model has better predicting power than classical logistic regression when the sample size is small. (C) 2019 Published by Elsevier B.V.

关键词： Robustness and sensitivity analysis Distributionally robust optimization Wasserstein metric semi-infinite programming Logistic regression

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Optimization-based global structural identifiability

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COMPUTERS & CHEMICAL ENGINEERING 2019年 128卷 417-420页

作者： Joy, Preet Djelassi, Hatim Mhamdi, Adel Mitsos, Alexander Rhein Westfal TH Aachen Aachener Verfahrenstech Proc Syst Engn SVT Forckenbeckstr 51 D-52074 Aachen Germany

Global structural identifiability determines if it is possible to uniquely estimate unknown parameters of a model from measurements. We consider two definitions for global structural identifiability - one proposed by Walter and Lecourtier in Mathematics and Computers in Simulation, 60: 472-482 (1982) and the other by Glad and Ljung in the Proceedings of the 29 th Conference on Decision and Control (1990). The two definitions appear distinct because of the role of the inputs. We present here a proof of the equivalence of the two definitions. We revisit the formulation of the optimization problem to analyze global structural identifiability proposed by Asprey and Mantalaris in IFAC Computer Application in Biotechnology (2001) and propose a modification to the problem formulation. We also demonstrate with the help of an elementary model that their solution algorithm can lead to erroneous conclusions. We further analyze the global structural identifiability of a simple model using the optimization-based approach. (C) 2019 Elsevier Ltd. All rights reserved.

关键词： Structural identifiability semi-infinite programming Global optimization

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Exact relaxations for parametric robust linear optimization problems

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OPERATIONS RESEARCH LETTERS 2019年第2期47卷 105-109页

作者： Thai Doan Chuong Ton Duc Thang Univ Optimizat & Applicat Res Grp Ho Chi Minh City Vietnam Ton Duc Thang Univ Fac Math & Stat Ho Chi Minh City Vietnam

We first show that the closedness of the characteristic cone of the constraint system of a parametric robust linear optimization problem is a necessary and sufficient condition for each robust linear program with the finite optimal value to admit exact semidefinite linear programming relaxations. We then provide the weakest regularity condition that guarantees exact second-order cone programming relaxations for parametric robust linear programs. (C) 2019 Elsevier B.V. All rights reserved.

关键词： semi-infinite programming Relaxation semidefinite linear program Robust linear programming Parametric optimization problem

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Nonsmooth semi-infinite programming problem using Limiting subdifferentials

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JOURNAL OF GLOBAL OPTIMIZATION 2012年第2期53卷 285-296页

作者： Mishra, S. K. Jaiswal, M. Le Thi, H. A. Banaras Hindu Univ Dept Math Varanasi 221005 Uttar Pradesh India Univ Paul Verlaine Metz UFR MIM Lab Theoret & Appl Comp Sci LITA Metz France

In this paper, we establish necessary and sufficient optimality conditions for nonsmooth semi-infinite programming problem using the powerful tool of limiting subdifferentials. We also formulate Wolfe and Mond-Weir type duals for nonsmooth semi-infinite programming problem and establish weak, strong and strict converse duality theorems for semi-infinite programming problem and the corresponding dual problems.

关键词： semi-infinite programming Nonsmooth optimization Sufficient optimality conditions Duality Generalized convexity

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The adaptive convexification algorithm for semi-infinite programming with arbitrary index sets

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MATHEMATICAL programming 2012年第1期136卷 183-207页

作者： Stein, Oliver Steuermann, Paul KIT Karlsruhe Germany

A numerical solution method for semi-infinite optimization problems with arbitrary, not necessarily box-shaped, index sets is presented. Following the ideas of Floudas and Stein (SIAM J Optim 18:1187-1208, 2007), convex relaxations of the lower level problem are adaptively constructed and then reformulated as mathematical programs with complementarity constraints and solved. Although the index set is arbitrary, this approximation produces feasible iterates for the original problem. The convex relaxations and needed parameters are constructed with ideas of the alpha BB method of global optimization and interval methods. It is shown that after finitely many steps an -stationary point of the original semi-infinite problem is reached. A numerical example illustrates the performance of the proposed method.

关键词： semi-infinite programming alpha BB Global optimization Convex optimization Mathematical programming with complementarity constraints Bilevel optimization

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Nonsmooth semi-infinite programming problem using Limiting subdifferentials

Nonsmooth semi-infinite programming problem using Limiting s...

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International Conference on Optimization and its Applications (ICOIA)

作者： Mishra, S. K. Jaiswal, M. Le Thi, H. A. Banaras Hindu Univ Dept Math Varanasi 221005 Uttar Pradesh India Univ Paul Verlaine Metz UFR MIM Lab Theoret & Appl Comp Sci LITA Metz France

关键词： semi-infinite programming Nonsmooth optimization Sufficient optimality conditions Duality Generalized convexity

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Set-membership nonlinear regression approach to parameter estimation

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JOURNAL OF PROCESS CONTROL 2018年 70卷 80-95页

作者： Peric, Nikola D. Paulen, Radoslav Villanueva, Mario E. Chachuat, Benoit Imperial Coll London Ctr Proc Syst Engn Dept Chem Engn London England ShanghaiTech Univ Sch Informat Sci & Technol Shanghai Peoples R China Slovak Univ Technol Bratislava Fac Chem & Food Technol Bratislava Slovakia

This paper introduces set-membership nonlinear regression (SMR), a new approach to nonlinear regression under uncertainty. The problem is to determine the subregion in parameter space enclosing all (global) solutions to a nonlinear regression problem in the presence of bounded uncertainty on the observed variables. Our focus is on nonlinear algebraic models. We investigate the connections of SMR with (i) the classical statistical inference methods, and (ii) the usual set-membership estimation approach where the model predictions are constrained within bounded measurement errors. We also develop a computational framework to describe tight enclosures of the SMR regions using semi-infinite programming and complete-search methods, in the form of likelihood contour and polyhedral enclosures. The case study of a parameter estimation problem in microbial growth is presented to illustrate various theoretical and computational aspects of the SMR approach. (C) 2018 The Author(s). Published by Elsevier Ltd.

关键词： Parameter estimation Nonlinear regression Set-membership estimation Statistical inference semi-infinite programming Complete-search methods

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Augmented Lagrangian functions for cone constrained optimization: the existence of global saddle points and exact penalty property

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JOURNAL OF GLOBAL OPTIMIZATION 2018年第2期71卷 237-296页

作者： Dolgopolik, M. V. St Petersburg State Univ St Petersburg Russia Russian Acad Sci Inst Problems Mech Engn St Petersburg Russia

In this article we present a general theory of augmented Lagrangian functions for cone constrained optimization problems that allows one to study almost all known augmented Lagrangians for these problems within a unified framework. We develop a new general method for proving the existence of global saddle points of augmented Lagrangian functions, called the localization principle. The localization principle unifies, generalizes and sharpens most of the known results on the existence of global saddle points, and, in essence, reduces the problem of the existence of global saddle points to a local analysis of optimality conditions. With the use of the localization principle we obtain first necessary and sufficient conditions for the existence of a global saddle point of an augmented Lagrangian for cone constrained minimax problems via both second and first order optimality conditions. In the second part of the paper, we present a general approach to the construction of globally exact augmented Lagrangian functions. The general approach developed in this paper allowed us not only to sharpen most of the existing results on globally exact augmented Lagrangians, but also to construct first globally exact augmented Lagrangian functions for equality constrained optimization problems, for nonlinear second order cone programs and for nonlinear semidefinite programs. These globally exact augmented Lagrangians can be utilized in order to design new superlinearly (or even quadratically) convergent optimization methods for cone constrained optimization problems.

关键词： Augmented Lagrangian Cone constrained optimization Localization principle Saddle point Second order cone semidefinite programming semi-infinite programming

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