Robust linear optimization under general norms
|
|
- Jeffrey Casey
- 6 years ago
- Views:
Transcription
1 Operations Research Letters 3 (004) Operations Research Letters Robust linear optimization under general norms Dimitris Bertsimas a; ;, Dessislava Pachamanova b, Melvyn Sim c a Sloan School of Management, Massachusetts Institute of Technology, Building E5-363, 50 Memorial Drive, Cambridge, MA , USA b Department of Mathematics and Sciences, Babson College, Forest Street, Babson Park, MA 0457, USA c NUS Business School, National University of Singapore, Business Link, Singapore 759, Singapore Received 8 August 003; received in revised form December 003; accepted 9 December 003 Abstract We explicitly characterize the robust counterpart of a linear programming problem with uncertainty set described by an arbitrary norm. Our approach encompasses several approaches from the literature and provides guarantees for constraint violation under probabilistic models that allow arbitrary dependencies in the distribution of the uncertain coecients. c 004 Elsevier B.V. All rights reserved. Keywords: Robust optimization; Stochastic programming; Linear programming; Norms; Dual norms; Constraint violation. Introduction Robust optimization is emerging as a leading methodology to address optimization problems under uncertainty. Applied to the linear programming problem max{c x Ãx 6 b; x P x } () with x R n, à R m n a matrix of uncertain coecients belonging to a known uncertainty set U, c R n and P x a given set representing the constraints involving certain coecients, the robust counterpart of Problem () is max{c x Ãx 6 b; x P x ; à U}: () Corresponding author. addresses: dbertsim@mit.edu (D. Bertsimas), dpachamanova@babson.edu (D. Pachamanova), dscsimm@nus.edu.sg (M. Sim). Research supported by the Singapore-MIT Alliance. An optimal solution x is robust with respect to any realization of the data, that is, it satises the constraints for any à U. The rst approach to modeling coecient uncertainty in the optimization literature was proposed by Soyster []. He considers problems of the kind max c x à j x j 6b; à j K j ;;:::;n;x 0 ; where x R n, c R n and b R m are data, and each column à j of the matrix à belongs to a convex set K j. Soyster shows that the problem is equivalent to a problem with linear constraints max c x A j x j 6 b; x 0 ; /$ - see front matter c 004 Elsevier B.V. All rights reserved. doi:0.06/j.orl
2 D. Bertsimas et al. / Operations Research Letters 3 (004) where the entries of the matrix A, a ij, satisfy a ij = sup A j K j (ã ij ). Some work followed Soyster s note (see, for example, [8]). However, the approaches suggested in this early literature solve a limited range of problems, are not easy to generalize, and are very conservative in the sense that they protect against the worst-case scenario. The interest in robust formulations in the optimization community was revived in the 990s. Ben-Tal and Nemirovski [ 4] introduced a number of important formulations and applications, and provided a detailed analysis of the robust optimization framework in linear programming and general convex programming. Independently, El Ghaoui et al. [9,0] derived similar results, and dealt with robust formulations of optimization problems by adapting robust control techniques. They showed that under the assumption that the coecient matrix data vary in ellipsoidal uncertainty sets, the robust counterparts of some important generic convex optimization problems (linear programming problems (LP), second-order cone problems (SOCP), semidenite programming problems (SDP)) are either exactly, or approximately tractable problems that are eciently solvable via interior point methods. However, under ellipsoidal uncertainty sets, the robust counterpart of an LP becomes an SOCP, of an SOCP becomes an SDP, while the robust counterpart of an SDP is NP-hard to solve. In other words, the diculty of the robust problem increases, as SDPs are numerically harder to solve than SOCPs, which in turn are harder to solve than LPs. Bertsimas and Sim [7] consider LPs such that each entry ã ij of à R m n is assumed to take values in the interval [ a ij ij ; a ij + ij ] and protect for the case that up to i of the uncertain coecients in constraint i, i= ;:::;m, can take their worst-case values at the same time. The parameter i controls the tradeo between robustness and optimality. Bertsimas and Sim show that the robust counterpart is still an LP. In this paper, we propose a framework for robust modeling of linear programming problems using uncertainty sets described by an arbitrary norm. We explicitly characterize the robust counterpart as a convex optimization problem that involves the dual norm of the given norm. Under a Euclidean norm we recover the second order cone formulation in [,,9,0], while under a particular D-norm we introduce we recover the linear programming formulation proposed in [7]. In this way, we shed some new light to the nature and structure of the robust counterpart of an LP. The paper is structured as follows. In Section, we review from [5] the robust counterpart of a linear programming problem when the deviations of the uncertain coecients lie in a convex set and characterize the robust counterpart of an LP when the uncertainty set is described by a general norm, as a convex optimization problem that involves the dual norm of the given norm. In Section 3, we show that by varying the norm used to dene the uncertainty set, we recover the second order cone formulation in [,,9,0], while under a particular D-norm we introduce we recover the linear programming formulation proposed in [7]. In Section 4, we provide guarantees for constraint violation under general probabilistic models that allow arbitrary dependencies in the distribution of the uncertain coecients. Notation. In this paper, lowercase boldface will be used to denote vectors, while uppercase boldface will denote matrices. Tilde (ã) will denote uncertain coef- cients, while check ( a) will be used for nominal values. à R m n will usually be the matrix of uncertain coecients in the linear programming problem, and vec(ã) R (m n) will denote the vector obtained by stacking its rows on top of one another.. Uncertainty sets dened by a norm In this section, we review from [5] the structure of the robust counterpart for uncertainty sets dened by general norms. These characterizations are used to develop the new characterizations in Section 3. Let S be a closed, bounded convex set and consider an uncertainty set in which the uncertain coecients are allowed to vary in such a way that the deviations from their nominal values fall in a convex set S U = {à (vec(ã) vec(a)) S}: The next theorem characterizes the robust counterpart. Theorem. Problem Ãx 6 b x P x à R m n such that (vec(ã) vec ( A)) S (3)
3 5 D. Bertsimas et al. / Operations Research Letters 3 (004) can be formulated as a i x + max y S {y x} 6 b i ; x P x : i=;:::;m (4) Proof. Clearly since S is compact, for each constraint i, ã ix 6 b i for all vec(ã) vec( A) S if and only if max {ã ix} 6 b i : {vec(ã) vec( A) S} Since max {ã ix} = max {(vec(ã)) x i } {vec(ã) vec( A) S} {vec(ã) vec( A) S} the theorem follows. = (vec( A)) x i + max {y x}; {y S} We next consider uncertainty sets that arise from the requirement that the distance (as measured by an arbitrary norm) between uncertain coecients and their nominal values is bounded. Specically, we consider an uncertainty set U given by U = {à M(vec(Ã) vec( A)) 6 }; (5) where M is an invertible matrix, 0 and a general norm. Given a norm x for a real space of vectors x, its dual norm induced over the dual space of linear functionals s is dened as follows: Denition (Dual Norm). s : = max { x 6} s x: (6) The next result is well known (see, for example, []). Proposition. (a) The dual norm of the dual norm is the original norm. (b) The dual norm of the L p norm =p : x p = x j p (7) is the L q norm s q with q =+=(p ). The next theorem derives the form of the robust counterpart, when the uncertainty set is given by Eq. (5). Theorem. Problem Ãx 6 b x P x à U = {à M(vec(Ã) vec( A)) 6 }; (8) where M is an invertible matrix, can be formulated as a i x + M xi 6 b i ; i=;:::;m x P x ; (9) where x i R (m n) is a vector that contains x R n in entries (i ) n+ through i n, and 0 everywhere else. Proof. Introducing y =(M(vec(Ã) vec( A)))=, we can write U as U = {y y 6 } and obtain max {ã ix} {vec(ã) U} = max {(vec(ã)) x i } {vec(ã) U} = max {(vec( A)) x i + (M y) x} {y y 6} = a x + max {y y 6} {y (M x)}: By Denition, the second term in the last expression is M x. The theorem follows by applying Theorem. In applications, [,,9,0] work with uncertainty sets given by the Euclidean norm, i.e., U = {à M(vec(Ã) vec( A)) 6 }; where denotes the Euclidean norm. Since the Euclidean norm is self dual, it follows that the robust counterpart is a second order cone problem. If the uncertainty set is described by either or
4 D. Bertsimas et al. / Operations Research Letters 3 (004) (these norms are dual to each other), then the resulting robust counterpart can be formulated as an LP. 3. The D-norm and its dual In this section, we show that the approach of [7] follows also from Theorem by simply considering a dierent norm, called the D-norm and its dual as opposed to the Euclidean norm considered in [,,9,0]. Furthermore, we show worst case bounds on the proximity of the D-norm to the Euclidean norm. Bertsimas and Sim [7] consider uncertainty sets given by &M(vec(Ã) vec( A)) & p 6 with p [;n] and for y R n = max {S {t} S N; S 6 p ;t N \S} y j +(p p ) y t : The fact that is indeed a norm, i.e., 0, &cy & p = c, =0 if and only y = 0, and &x + y & p 6 &x & p + follows easily. Specically [7] considers constraint-wise uncertainty sets, M a diagonal matrix containing the inverses of the ranges of coecient variation, and =. We next derive the dual norm. Proposition. The dual norm of the norm & & p is given by &s & p = max( s ; s =p): Proof. The norm can be written as = max u j y j u j 6 p; 0 6 u j 6 = min pr + t j r + t j y j ;t j 0; r 0; where the second equality follows by linear programming strong duality. Thus, 6 if and only if pr + t j 6 ;r+ t j y j ;t j 0; r 0 (0) is feasible. The dual norm &s & p is given by &s & p = & max s y: y & p 6 From Eq. (0) we have that &s & p = max s y pr + t j 6 ; y j t j r 6 0; y j t j r 6 0; r 0;t j 0; Applying LP duality again we obtain &s & p = min p u j Thus, &s & p = min u j v j 0; u j v j = s j ; 0;u j ;v j 0; j =;:::;n: v j 0; j =;:::;n: s j ; s j =p; and hence, &s & p = max( s ; s =p): Combining Theorem and Proposition leads to an LP formulation of the robust counterpart of the
5 54 D. Bertsimas et al. / Operations Research Letters 3 (004) uncertainty set in [7]. We thus observe that Theorem provides a unied treatment of the approaches in [,,7,9,0]. 3.. Comparison with the Euclidean norm Since uncertainty sets in the literature have been described using the Euclidean and the D-norm it is of interest to understand the proximity between these two norms. Proposition 3. For every y R n { min ; } p & y & p 6 6 p +(p p ) n y { } min p ; & y n & { } p n 6 6 max y p ; : Proof. We nd a lower bound on = y by solving the following nonlinear optimization problem: max xj () j N &x & p =: Let S = {;:::; p }, t = p +. We can impose nonnegativity constraints on x and the constraints that x j x t ; j S and x j 6 x t ; j N \ S, without aecting the objective value of the Problem (). Observing that the objective can never decrease if we let x j = x t ; j N\S, we reduce () to the following problem: max xj +(n p )xt x j +(p p )x t =; x j x t j S; x t 0: () Since we are maximizing a convex function over a polytope, there exist an extreme point optimal solution to Problem (). There are S + inequality constraints. Out of those, S need to be tight to establish an extreme point solution. The S + extreme points can be found to be x k = e k k S; (3) x S + = e; (4) p where e is the vector of ones and e k is the unit vector with the kth element equals one, and the rest equal zero. Clearly, substituting all possible solutions, Problem () yields the optimum value of max{; n=p }. Taking the square root, the inequality y 6 max{; n=p} follows. Similarly, in order to derive an upper bound of = y, we solve the following problem: min xj j N (5) &x & p =: Using the same partition of the solution an before, and observing that the objective can never increase with x j =0; j N \ S \{t}, we reduce Problem (5) to the following problem: min xj + xt x j +(p p )x t =; x j x t j S; x t 0: (6) An optimal solution of Problem (6) can be found using the KKT conditions: p +(p p ) if j S; x j = p p p +(p p ) if j = t; 0 otherwise: Substituting, the optimal objective value of Problem (6) is ( p +(p p ) ). Hence, taking the square root, we establish the inequality 6 p +(p p ) y. Since 6 y 6 n; 6 y 6 ; y n y
6 D. Bertsimas et al. / Operations Research Letters 3 (004) and { } &y & p = max y p ; y the bounds follow. 4. Probabilistic guarantees In this section, we derive probabilistic guarantees on the feasibility of an optimal robust solution when the uncertainty set U is described by a general norm with a dual norm. We assume that the matrix à has an arbitrary (and unknown) distribution, but with known expected value A R m n and known covariance matrix R (m n) (m n). Note that we allow arbitrary dependencies on Ã. We dene M = =, which is a symmetric matrix. Let x R n be an optimal solution to Problem (9). Recall that xi R (m n) denotes the vector containing x in entries (i ) n through i n, and 0 everywhere else. Theorem 3. (a) The probability that x satises constraint i for any realization of the uncertain matrix à is P(ã ix 6 b i ) =P((vec(Ã)) x i 6 b i ) + ( = xi = = xi ) : (7) (b) If the norm used in U is the D-norm & & p, then P(ã ix 6 b i ) { + min p ; n }: (8) (c) If the norm used in U is the dual D-norm & & p, then P(ã ix 6 b i ) }: (9) + min {; p n (d) If the norm used in U is the Euclidean norm, then P(ã ix 6 b i ) + : (0) Proof. Since an optimal robust solution satises (vec( A)) xi + = xi 6 b i ; we obtain that P((vec(Ã)) xi b i )6P((vec(Ã)) xi (vec( A)) xi + = xi ): () Bertsimas and Popescu [6] show that if S is a convex set, and X is a vector of random variables with mean X and covariance matrix, then P( X S) 6 +d ; () where d = inf ( X X) ( X X): X S We consider the convex set S i ={vec(ã) (vec(ã)) xi (vec( A)) xi + = xi }. In our case, d i = inf (vec(ã) vec( A)) (vec(ã) vec(ã) S i vec( A)): Applying the KKT conditions for this optimization problem we obtain an optimal solution ( ) vec( A) = xi + = xi xi with ( ) d = = xi = xi : Applying (), Eq. (7) follows. (b) If the norm used in the uncertainty set U is the D-norm, then by applying Proposition 3, we obtain Eq. (8). (c) If the norm used in the uncertainty set U is the dual D-norm, then by applying Proposition 3, we obtain Eq. (9). (d) If the norm used in the uncertainty set U is the Euclidean norm, then Eq. (0) follows immediately from Eq. (7). Under the stronger assumption that the data in each constraint are independently distributed according to a symmetric distribution [7] proves stronger bounds. In contrast the bounds in Theorem 3 are weaker, but
7 56 D. Bertsimas et al. / Operations Research Letters 3 (004) have wider applicability as they include arbitrary dependencies. References [] A. Ben-Tal, A. Nemirovski, Robust convex optimization, Math. Oper. Res. 3 (4) (998) [] A. Ben-Tal, A. Nemirovski, Robust solutions of uncertain linear programs, Oper. Res. Lett. 5 () (999) 3. [3] A. Ben-Tal, A. Nemirovski, Robust solutions of linear programming problems contaminated with uncertain data, Math. Program. A 88 (000) [4] A. Ben-Tal, A. Nemirovski, On polyhedral approximations of the second-order Cone, Math. Oper. Res. 6 () (00) [5] A. Ben-Tal and A. Nemirovski, Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications, MPR-SIAM Series on Optimization, SIAM, Philadelphia, 00. [6] D. Bertsimas, I. Popescu, Optimal inequalities in probability theory: a convex optimization approach, in SIAM J. Optim. 004, to appear. [7] D. Bertsimas, M. Sim, The price of robustness, Oper. Res. 5 () (004) [8] J.E. Falk, Exact solutions of inexact linear programs, Oper. Res. 4 (976) [9] L. El Ghaoui, H. Lebret, Robust solutions to least-squares problems with uncertain data, SIAM J Matrix Anal. Appl. 8 (4) (997) [0] L. El Ghaoui, F. Oustry, H. Lebret, Robust solutions to uncertain semidenite programs, SIAM J. Optim. 9 () (998) [] P.D. Lax, Linear Algebra, Wiley, New York, 997. [] A.L. Soyster, Convex programming with set-inclusive constraints and applications to inexact linear programming, Oper. Res (973)
Robust portfolio selection under norm uncertainty
Wang and Cheng Journal of Inequalities and Applications (2016) 2016:164 DOI 10.1186/s13660-016-1102-4 R E S E A R C H Open Access Robust portfolio selection under norm uncertainty Lei Wang 1 and Xi Cheng
More informationHandout 8: Dealing with Data Uncertainty
MFE 5100: Optimization 2015 16 First Term Handout 8: Dealing with Data Uncertainty Instructor: Anthony Man Cho So December 1, 2015 1 Introduction Conic linear programming CLP, and in particular, semidefinite
More informationRobust Farkas Lemma for Uncertain Linear Systems with Applications
Robust Farkas Lemma for Uncertain Linear Systems with Applications V. Jeyakumar and G. Li Revised Version: July 8, 2010 Abstract We present a robust Farkas lemma, which provides a new generalization of
More informationTractable Approximations to Robust Conic Optimization Problems
Math. Program., Ser. B 107, 5 36 2006) Digital Object Identifier DOI) 10.1007/s10107-005-0677-1 Dimitris Bertsimas Melvyn Sim Tractable Approximations to Robust Conic Optimization Problems Received: April
More informationSemidefinite and Second Order Cone Programming Seminar Fall 2012 Project: Robust Optimization and its Application of Robust Portfolio Optimization
Semidefinite and Second Order Cone Programming Seminar Fall 2012 Project: Robust Optimization and its Application of Robust Portfolio Optimization Instructor: Farid Alizadeh Author: Ai Kagawa 12/12/2012
More information1 Robust optimization
ORF 523 Lecture 16 Princeton University Instructor: A.A. Ahmadi Scribe: G. Hall Any typos should be emailed to a a a@princeton.edu. In this lecture, we give a brief introduction to robust optimization
More informationLIGHT ROBUSTNESS. Matteo Fischetti, Michele Monaci. DEI, University of Padova. 1st ARRIVAL Annual Workshop and Review Meeting, Utrecht, April 19, 2007
LIGHT ROBUSTNESS Matteo Fischetti, Michele Monaci DEI, University of Padova 1st ARRIVAL Annual Workshop and Review Meeting, Utrecht, April 19, 2007 Work supported by the Future and Emerging Technologies
More informationOperations Research Letters
Operations Research Letters 37 (2009) 1 6 Contents lists available at ScienceDirect Operations Research Letters journal homepage: www.elsevier.com/locate/orl Duality in robust optimization: Primal worst
More informationHandout 6: Some Applications of Conic Linear Programming
ENGG 550: Foundations of Optimization 08 9 First Term Handout 6: Some Applications of Conic Linear Programming Instructor: Anthony Man Cho So November, 08 Introduction Conic linear programming CLP, and
More informationThe Value of Adaptability
The Value of Adaptability Dimitris Bertsimas Constantine Caramanis September 30, 2005 Abstract We consider linear optimization problems with deterministic parameter uncertainty. We consider a departure
More informationRobust discrete optimization and network flows
Math. Program., Ser. B 98: 49 71 2003) Digital Object Identifier DOI) 10.1007/s10107-003-0396-4 Dimitris Bertsimas Melvyn Sim Robust discrete optimization and network flows Received: January 1, 2002 /
More informationOn the Power of Robust Solutions in Two-Stage Stochastic and Adaptive Optimization Problems
MATHEMATICS OF OPERATIONS RESEARCH Vol. 35, No., May 010, pp. 84 305 issn 0364-765X eissn 156-5471 10 350 084 informs doi 10.187/moor.1090.0440 010 INFORMS On the Power of Robust Solutions in Two-Stage
More informationRobust Scheduling with Logic-Based Benders Decomposition
Robust Scheduling with Logic-Based Benders Decomposition Elvin Çoban and Aliza Heching and J N Hooker and Alan Scheller-Wolf Abstract We study project scheduling at a large IT services delivery center
More informationInterval solutions for interval algebraic equations
Mathematics and Computers in Simulation 66 (2004) 207 217 Interval solutions for interval algebraic equations B.T. Polyak, S.A. Nazin Institute of Control Sciences, Russian Academy of Sciences, 65 Profsoyuznaya
More informationStrong Duality in Robust Semi-Definite Linear Programming under Data Uncertainty
Strong Duality in Robust Semi-Definite Linear Programming under Data Uncertainty V. Jeyakumar and G. Y. Li March 1, 2012 Abstract This paper develops the deterministic approach to duality for semi-definite
More informationPareto Efficiency in Robust Optimization
Pareto Efficiency in Robust Optimization Dan Iancu Graduate School of Business Stanford University joint work with Nikolaos Trichakis (HBS) 1/26 Classical Robust Optimization Typical linear optimization
More information1 Introduction Semidenite programming (SDP) has been an active research area following the seminal work of Nesterov and Nemirovski [9] see also Alizad
Quadratic Maximization and Semidenite Relaxation Shuzhong Zhang Econometric Institute Erasmus University P.O. Box 1738 3000 DR Rotterdam The Netherlands email: zhang@few.eur.nl fax: +31-10-408916 August,
More informationminimize x x2 2 x 1x 2 x 1 subject to x 1 +2x 2 u 1 x 1 4x 2 u 2, 5x 1 +76x 2 1,
4 Duality 4.1 Numerical perturbation analysis example. Consider the quadratic program with variables x 1, x 2, and parameters u 1, u 2. minimize x 2 1 +2x2 2 x 1x 2 x 1 subject to x 1 +2x 2 u 1 x 1 4x
More informationLecture 6: Conic Optimization September 8
IE 598: Big Data Optimization Fall 2016 Lecture 6: Conic Optimization September 8 Lecturer: Niao He Scriber: Juan Xu Overview In this lecture, we finish up our previous discussion on optimality conditions
More informationMulti-Range Robust Optimization vs Stochastic Programming in Prioritizing Project Selection
Multi-Range Robust Optimization vs Stochastic Programming in Prioritizing Project Selection Ruken Düzgün Aurélie Thiele July 2012 Abstract This paper describes a multi-range robust optimization approach
More informationRobust Optimization for Risk Control in Enterprise-wide Optimization
Robust Optimization for Risk Control in Enterprise-wide Optimization Juan Pablo Vielma Department of Industrial Engineering University of Pittsburgh EWO Seminar, 011 Pittsburgh, PA Uncertainty in Optimization
More informationDistributionally Robust Optimization with ROME (part 1)
Distributionally Robust Optimization with ROME (part 1) Joel Goh Melvyn Sim Department of Decision Sciences NUS Business School, Singapore 18 Jun 2009 NUS Business School Guest Lecture J. Goh, M. Sim (NUS)
More informationROBUST CONVEX OPTIMIZATION
ROBUST CONVEX OPTIMIZATION A. BEN-TAL AND A. NEMIROVSKI We study convex optimization problems for which the data is not specified exactly and it is only known to belong to a given uncertainty set U, yet
More informationA Geometric Characterization of the Power of Finite Adaptability in Multistage Stochastic and Adaptive Optimization
MATHEMATICS OF OPERATIONS RESEARCH Vol. 36, No., February 20, pp. 24 54 issn 0364-765X eissn 526-547 360 0024 informs doi 0.287/moor.0.0482 20 INFORMS A Geometric Characterization of the Power of Finite
More informationDistributionally Robust Discrete Optimization with Entropic Value-at-Risk
Distributionally Robust Discrete Optimization with Entropic Value-at-Risk Daniel Zhuoyu Long Department of SEEM, The Chinese University of Hong Kong, zylong@se.cuhk.edu.hk Jin Qi NUS Business School, National
More informationRobust Combinatorial Optimization under Budgeted-Ellipsoidal Uncertainty
EURO Journal on Computational Optimization manuscript No. (will be inserted by the editor) Robust Combinatorial Optimization under Budgeted-Ellipsoidal Uncertainty Jannis Kurtz Received: date / Accepted:
More informationOn the Power of Robust Solutions in Two-Stage Stochastic and Adaptive Optimization Problems
MATHEMATICS OF OPERATIONS RESEARCH Vol. xx, No. x, Xxxxxxx 00x, pp. xxx xxx ISSN 0364-765X EISSN 156-5471 0x xx0x 0xxx informs DOI 10.187/moor.xxxx.xxxx c 00x INFORMS On the Power of Robust Solutions in
More informationAbstract. SOCPs remain SOCPs and robust SDPs remain SDPs moreover, when the data entries are
Robust Conic Optimization Dimitris Bertsimas Melvyn Sim y March 4 Abstract In earlier proposals, the robust counterpart of conic optimization problems exhibits a lateral increase in complexity, i.e., robust
More informationConvex Optimization in Classification Problems
New Trends in Optimization and Computational Algorithms December 9 13, 2001 Convex Optimization in Classification Problems Laurent El Ghaoui Department of EECS, UC Berkeley elghaoui@eecs.berkeley.edu 1
More informationMin-max-min robustness: a new approach to combinatorial optimization under uncertainty based on multiple solutions 1
Min-max- robustness: a new approach to combinatorial optimization under uncertainty based on multiple solutions 1 Christoph Buchheim, Jannis Kurtz 2 Faultät Mathemati, Technische Universität Dortmund Vogelpothsweg
More informationRobust conic quadratic programming with ellipsoidal uncertainties
Robust conic quadratic programming with ellipsoidal uncertainties Roland Hildebrand (LJK Grenoble 1 / CNRS, Grenoble) KTH, Stockholm; November 13, 2008 1 Uncertain conic programs min x c, x : Ax + b K
More informationAssignment 1: From the Definition of Convexity to Helley Theorem
Assignment 1: From the Definition of Convexity to Helley Theorem Exercise 1 Mark in the following list the sets which are convex: 1. {x R 2 : x 1 + i 2 x 2 1, i = 1,..., 10} 2. {x R 2 : x 2 1 + 2ix 1x
More informationAbsolute value equations
Linear Algebra and its Applications 419 (2006) 359 367 www.elsevier.com/locate/laa Absolute value equations O.L. Mangasarian, R.R. Meyer Computer Sciences Department, University of Wisconsin, 1210 West
More informationLecture: Cone programming. Approximating the Lorentz cone.
Strong relaxations for discrete optimization problems 10/05/16 Lecture: Cone programming. Approximating the Lorentz cone. Lecturer: Yuri Faenza Scribes: Igor Malinović 1 Introduction Cone programming is
More informationDistributionally Robust Convex Optimization
Distributionally Robust Convex Optimization Wolfram Wiesemann 1, Daniel Kuhn 1, and Melvyn Sim 2 1 Department of Computing, Imperial College London, United Kingdom 2 Department of Decision Sciences, National
More informationTrust Region Problems with Linear Inequality Constraints: Exact SDP Relaxation, Global Optimality and Robust Optimization
Trust Region Problems with Linear Inequality Constraints: Exact SDP Relaxation, Global Optimality and Robust Optimization V. Jeyakumar and G. Y. Li Revised Version: September 11, 2013 Abstract The trust-region
More informationSelected Examples of CONIC DUALITY AT WORK Robust Linear Optimization Synthesis of Linear Controllers Matrix Cube Theorem A.
. Selected Examples of CONIC DUALITY AT WORK Robust Linear Optimization Synthesis of Linear Controllers Matrix Cube Theorem A. Nemirovski Arkadi.Nemirovski@isye.gatech.edu Linear Optimization Problem,
More informationA Hierarchy of Polyhedral Approximations of Robust Semidefinite Programs
A Hierarchy of Polyhedral Approximations of Robust Semidefinite Programs Raphael Louca Eilyan Bitar Abstract Robust semidefinite programs are NP-hard in general In contrast, robust linear programs admit
More informationA Robust Optimization Perspective on Stochastic Programming
A Robust Optimization Perspective on Stochastic Programming Xin Chen, Melvyn Sim and Peng Sun Submitted: December 004 Revised: May 16, 006 Abstract In this paper, we introduce an approach for constructing
More informationConvex Optimization. (EE227A: UC Berkeley) Lecture 6. Suvrit Sra. (Conic optimization) 07 Feb, 2013
Convex Optimization (EE227A: UC Berkeley) Lecture 6 (Conic optimization) 07 Feb, 2013 Suvrit Sra Organizational Info Quiz coming up on 19th Feb. Project teams by 19th Feb Good if you can mix your research
More informationRobust combinatorial optimization with variable budgeted uncertainty
Noname manuscript No. (will be inserted by the editor) Robust combinatorial optimization with variable budgeted uncertainty Michael Poss Received: date / Accepted: date Abstract We introduce a new model
More informationRobust economic-statistical design of multivariate exponentially weighted moving average control chart under uncertainty with interval data
Scientia Iranica E (2015) 22(3), 1189{1202 Sharif University of Technology Scientia Iranica Transactions E: Industrial Engineering www.scientiairanica.com Robust economic-statistical design of multivariate
More informationRobust Dual-Response Optimization
Yanıkoğlu, den Hertog, and Kleijnen Robust Dual-Response Optimization 29 May 1 June 1 / 24 Robust Dual-Response Optimization İhsan Yanıkoğlu, Dick den Hertog, Jack P.C. Kleijnen Özyeğin University, İstanbul,
More informationE5295/5B5749 Convex optimization with engineering applications. Lecture 5. Convex programming and semidefinite programming
E5295/5B5749 Convex optimization with engineering applications Lecture 5 Convex programming and semidefinite programming A. Forsgren, KTH 1 Lecture 5 Convex optimization 2006/2007 Convex quadratic program
More informationTheory and applications of Robust Optimization
Theory and applications of Robust Optimization Dimitris Bertsimas, David B. Brown, Constantine Caramanis May 31, 2007 Abstract In this paper we survey the primary research, both theoretical and applied,
More informationA Robust von Neumann Minimax Theorem for Zero-Sum Games under Bounded Payoff Uncertainty
A Robust von Neumann Minimax Theorem for Zero-Sum Games under Bounded Payoff Uncertainty V. Jeyakumar, G.Y. Li and G. M. Lee Revised Version: January 20, 2011 Abstract The celebrated von Neumann minimax
More informationA Geometric Characterization of the Power of Finite Adaptability in Multi-stage Stochastic and Adaptive Optimization
A Geometric Characterization of the Power of Finite Adaptability in Multi-stage Stochastic and Adaptive Optimization Dimitris Bertsimas Sloan School of Management and Operations Research Center, Massachusetts
More informationDistributionally Robust Convex Optimization
Submitted to Operations Research manuscript OPRE-2013-02-060 Authors are encouraged to submit new papers to INFORMS journals by means of a style file template, which includes the journal title. However,
More informationDeterminant maximization with linear. S. Boyd, L. Vandenberghe, S.-P. Wu. Information Systems Laboratory. Stanford University
Determinant maximization with linear matrix inequality constraints S. Boyd, L. Vandenberghe, S.-P. Wu Information Systems Laboratory Stanford University SCCM Seminar 5 February 1996 1 MAXDET problem denition
More informationLecture 1. 1 Conic programming. MA 796S: Convex Optimization and Interior Point Methods October 8, Consider the conic program. min.
MA 796S: Convex Optimization and Interior Point Methods October 8, 2007 Lecture 1 Lecturer: Kartik Sivaramakrishnan Scribe: Kartik Sivaramakrishnan 1 Conic programming Consider the conic program min s.t.
More informationRobust l 1 and l Solutions of Linear Inequalities
Available at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 1932-9466 Applications and Applied Mathematics: An International Journal (AAM) Vol. 6, Issue 2 (December 2011), pp. 522-528 Robust l 1 and l Solutions
More informationCSC Linear Programming and Combinatorial Optimization Lecture 10: Semidefinite Programming
CSC2411 - Linear Programming and Combinatorial Optimization Lecture 10: Semidefinite Programming Notes taken by Mike Jamieson March 28, 2005 Summary: In this lecture, we introduce semidefinite programming
More informationRestricted robust uniform matroid maximization under interval uncertainty
Math. Program., Ser. A (2007) 110:431 441 DOI 10.1007/s10107-006-0008-1 FULL LENGTH PAPER Restricted robust uniform matroid maximization under interval uncertainty H. Yaman O. E. Karaşan M. Ç. Pınar Received:
More informationU.C. Berkeley CS294: Beyond Worst-Case Analysis Handout 12 Luca Trevisan October 3, 2017
U.C. Berkeley CS94: Beyond Worst-Case Analysis Handout 1 Luca Trevisan October 3, 017 Scribed by Maxim Rabinovich Lecture 1 In which we begin to prove that the SDP relaxation exactly recovers communities
More informationThe moment-lp and moment-sos approaches
The moment-lp and moment-sos approaches LAAS-CNRS and Institute of Mathematics, Toulouse, France CIRM, November 2013 Semidefinite Programming Why polynomial optimization? LP- and SDP- CERTIFICATES of POSITIVITY
More informationMIT Algebraic techniques and semidefinite optimization February 14, Lecture 3
MI 6.97 Algebraic techniques and semidefinite optimization February 4, 6 Lecture 3 Lecturer: Pablo A. Parrilo Scribe: Pablo A. Parrilo In this lecture, we will discuss one of the most important applications
More informationConvex relaxation. In example below, we have N = 6, and the cut we are considering
Convex relaxation The art and science of convex relaxation revolves around taking a non-convex problem that you want to solve, and replacing it with a convex problem which you can actually solve the solution
More informationApplications of Robust Optimization in Signal Processing: Beamforming and Power Control Fall 2012
Applications of Robust Optimization in Signal Processing: Beamforg and Power Control Fall 2012 Instructor: Farid Alizadeh Scribe: Shunqiao Sun 12/09/2012 1 Overview In this presentation, we study the applications
More informationIntroduction to Mathematical Programming IE406. Lecture 10. Dr. Ted Ralphs
Introduction to Mathematical Programming IE406 Lecture 10 Dr. Ted Ralphs IE406 Lecture 10 1 Reading for This Lecture Bertsimas 4.1-4.3 IE406 Lecture 10 2 Duality Theory: Motivation Consider the following
More informationA Linear Storage-Retrieval Policy for Robust Warehouse Management
A Linear Storage-Retrieval Policy for Robust Warehouse Management Marcus Ang Yun Fong Lim Melvyn Sim Singapore-MIT Alliance, National University of Singapore, 4 Engineering Drive 3, Singapore 117576, Singapore
More informationAn Alternative Proof of Primitivity of Indecomposable Nonnegative Matrices with a Positive Trace
An Alternative Proof of Primitivity of Indecomposable Nonnegative Matrices with a Positive Trace Takao Fujimoto Abstract. This research memorandum is aimed at presenting an alternative proof to a well
More informationLec12p1, ORF363/COS323. The idea behind duality. This lecture
Lec12 Page 1 Lec12p1, ORF363/COS323 This lecture Linear programming duality + robust linear programming Intuition behind the derivation of the dual Weak and strong duality theorems Max-flow=Min-cut Primal/dual
More informationStrong Formulations of Robust Mixed 0 1 Programming
Math. Program., Ser. B 108, 235 250 (2006) Digital Object Identifier (DOI) 10.1007/s10107-006-0709-5 Alper Atamtürk Strong Formulations of Robust Mixed 0 1 Programming Received: January 27, 2004 / Accepted:
More informationConvex relaxation. In example below, we have N = 6, and the cut we are considering
Convex relaxation The art and science of convex relaxation revolves around taking a non-convex problem that you want to solve, and replacing it with a convex problem which you can actually solve the solution
More informationSemidefinite Programming Basics and Applications
Semidefinite Programming Basics and Applications Ray Pörn, principal lecturer Åbo Akademi University Novia University of Applied Sciences Content What is semidefinite programming (SDP)? How to represent
More informationOn the Relation Between Flexibility Analysis and Robust Optimization for Linear Systems
On the Relation Between Flexibility Analysis and Robust Optimization for Linear Systems Qi Zhang a, Ricardo M. Lima b, Ignacio E. Grossmann a, a Center for Advanced Process Decision-making, Department
More informationA Hierarchy of Suboptimal Policies for the Multi-period, Multi-echelon, Robust Inventory Problem
A Hierarchy of Suboptimal Policies for the Multi-period, Multi-echelon, Robust Inventory Problem Dimitris J. Bertsimas Dan A. Iancu Pablo A. Parrilo Sloan School of Management and Operations Research Center,
More informationA Linear Decision-Based Approximation Approach to Stochastic Programming
OPERATIONS RESEARCH Vol. 56, No. 2, March April 2008, pp. 344 357 issn 0030-364X eissn 526-5463 08 5602 0344 informs doi 0.287/opre.070.0457 2008 INFORMS A Linear Decision-Based Approximation Approach
More informationCSCI 1951-G Optimization Methods in Finance Part 10: Conic Optimization
CSCI 1951-G Optimization Methods in Finance Part 10: Conic Optimization April 6, 2018 1 / 34 This material is covered in the textbook, Chapters 9 and 10. Some of the materials are taken from it. Some of
More informationNEW ROBUST UNSUPERVISED SUPPORT VECTOR MACHINES
J Syst Sci Complex () 24: 466 476 NEW ROBUST UNSUPERVISED SUPPORT VECTOR MACHINES Kun ZHAO Mingyu ZHANG Naiyang DENG DOI:.7/s424--82-8 Received: March 8 / Revised: 5 February 9 c The Editorial Office of
More informationRobust Fisher Discriminant Analysis
Robust Fisher Discriminant Analysis Seung-Jean Kim Alessandro Magnani Stephen P. Boyd Information Systems Laboratory Electrical Engineering Department, Stanford University Stanford, CA 94305-9510 sjkim@stanford.edu
More informationA new primal-dual path-following method for convex quadratic programming
Volume 5, N., pp. 97 0, 006 Copyright 006 SBMAC ISSN 00-805 www.scielo.br/cam A new primal-dual path-following method for convex quadratic programming MOHAMED ACHACHE Département de Mathématiques, Faculté
More informationLifting for conic mixed-integer programming
Math. Program., Ser. A DOI 1.17/s117-9-282-9 FULL LENGTH PAPER Lifting for conic mixed-integer programming Alper Atamtürk Vishnu Narayanan Received: 13 March 28 / Accepted: 28 January 29 The Author(s)
More informationRobust-to-Dynamics Linear Programming
Robust-to-Dynamics Linear Programg Amir Ali Ahmad and Oktay Günlük Abstract We consider a class of robust optimization problems that we call robust-to-dynamics optimization (RDO) The input to an RDO problem
More informationOn the Chvatál-Complexity of Binary Knapsack Problems. Gergely Kovács 1 Béla Vizvári College for Modern Business Studies, Hungary
On the Chvatál-Complexity of Binary Knapsack Problems Gergely Kovács 1 Béla Vizvári 2 1 College for Modern Business Studies, Hungary 2 Eastern Mediterranean University, TRNC 2009. 1 Chvátal Cut and Complexity
More informationORIE 6300 Mathematical Programming I August 25, Lecture 2
ORIE 6300 Mathematical Programming I August 25, 2016 Lecturer: Damek Davis Lecture 2 Scribe: Johan Bjorck Last time, we considered the dual of linear programs in our basic form: max(c T x : Ax b). We also
More informationEE 227A: Convex Optimization and Applications April 24, 2008
EE 227A: Convex Optimization and Applications April 24, 2008 Lecture 24: Robust Optimization: Chance Constraints Lecturer: Laurent El Ghaoui Reading assignment: Chapter 2 of the book on Robust Optimization
More informationAgenda. 1 Cone programming. 2 Convex cones. 3 Generalized inequalities. 4 Linear programming (LP) 5 Second-order cone programming (SOCP)
Agenda 1 Cone programming 2 Convex cones 3 Generalized inequalities 4 Linear programming (LP) 5 Second-order cone programming (SOCP) 6 Semidefinite programming (SDP) 7 Examples Optimization problem in
More informationTheory and Applications of Robust Optimization
Theory and Applications of Robust Optimization The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation As Published Publisher Bertsimas,
More informationApproximation of Ellipsoids Using Bounded Uncertainty Sets
Approximation of Ellipsoids Using Bounded Uncertainty Sets André Chassein Fachbereich Mathematik, Technische Universität Kaiserslautern, Germany Abstract In this paper, we discuss the problem of approximating
More informationLecture: Introduction to LP, SDP and SOCP
Lecture: Introduction to LP, SDP and SOCP Zaiwen Wen Beijing International Center For Mathematical Research Peking University http://bicmr.pku.edu.cn/~wenzw/bigdata2015.html wenzw@pku.edu.cn Acknowledgement:
More information15-780: LinearProgramming
15-780: LinearProgramming J. Zico Kolter February 1-3, 2016 1 Outline Introduction Some linear algebra review Linear programming Simplex algorithm Duality and dual simplex 2 Outline Introduction Some linear
More information3. Linear Programming and Polyhedral Combinatorics
Massachusetts Institute of Technology 18.433: Combinatorial Optimization Michel X. Goemans February 28th, 2013 3. Linear Programming and Polyhedral Combinatorics Summary of what was seen in the introductory
More information1 Outline Part I: Linear Programming (LP) Interior-Point Approach 1. Simplex Approach Comparison Part II: Semidenite Programming (SDP) Concludin
Sensitivity Analysis in LP and SDP Using Interior-Point Methods E. Alper Yldrm School of Operations Research and Industrial Engineering Cornell University Ithaca, NY joint with Michael J. Todd INFORMS
More informationCanonical Problem Forms. Ryan Tibshirani Convex Optimization
Canonical Problem Forms Ryan Tibshirani Convex Optimization 10-725 Last time: optimization basics Optimization terology (e.g., criterion, constraints, feasible points, solutions) Properties and first-order
More information15. Conic optimization
L. Vandenberghe EE236C (Spring 216) 15. Conic optimization conic linear program examples modeling duality 15-1 Generalized (conic) inequalities Conic inequality: a constraint x K where K is a convex cone
More informationRelaxations and Randomized Methods for Nonconvex QCQPs
Relaxations and Randomized Methods for Nonconvex QCQPs Alexandre d Aspremont, Stephen Boyd EE392o, Stanford University Autumn, 2003 Introduction While some special classes of nonconvex problems can be
More informationAcyclic Semidefinite Approximations of Quadratically Constrained Quadratic Programs
2015 American Control Conference Palmer House Hilton July 1-3, 2015. Chicago, IL, USA Acyclic Semidefinite Approximations of Quadratically Constrained Quadratic Programs Raphael Louca and Eilyan Bitar
More informationOn duality gap in linear conic problems
On duality gap in linear conic problems C. Zălinescu Abstract In their paper Duality of linear conic problems A. Shapiro and A. Nemirovski considered two possible properties (A) and (B) for dual linear
More informationRobust and Stochastic Optimization Notes. Kevin Kircher, Cornell MAE
Robust and Stochastic Optimization Notes Kevin Kircher, Cornell MAE These are partial notes from ECE 6990, Robust and Stochastic Optimization, as taught by Prof. Eilyan Bitar at Cornell University in the
More informationA TOUR OF LINEAR ALGEBRA FOR JDEP 384H
A TOUR OF LINEAR ALGEBRA FOR JDEP 384H Contents Solving Systems 1 Matrix Arithmetic 3 The Basic Rules of Matrix Arithmetic 4 Norms and Dot Products 5 Norms 5 Dot Products 6 Linear Programming 7 Eigenvectors
More informationy Ray of Half-line or ray through in the direction of y
Chapter LINEAR COMPLEMENTARITY PROBLEM, ITS GEOMETRY, AND APPLICATIONS. THE LINEAR COMPLEMENTARITY PROBLEM AND ITS GEOMETRY The Linear Complementarity Problem (abbreviated as LCP) is a general problem
More informationShort Course Robust Optimization and Machine Learning. Lecture 6: Robust Optimization in Machine Learning
Short Course Robust Optimization and Machine Machine Lecture 6: Robust Optimization in Machine Laurent El Ghaoui EECS and IEOR Departments UC Berkeley Spring seminar TRANSP-OR, Zinal, Jan. 16-19, 2012
More informationNP-hardness of the stable matrix in unit interval family problem in discrete time
Systems & Control Letters 42 21 261 265 www.elsevier.com/locate/sysconle NP-hardness of the stable matrix in unit interval family problem in discrete time Alejandra Mercado, K.J. Ray Liu Electrical and
More informationRobust optimal solutions in interval linear programming with forall-exists quantifiers
Robust optimal solutions in interval linear programming with forall-exists quantifiers Milan Hladík October 5, 2018 arxiv:1403.7427v1 [math.oc] 28 Mar 2014 Abstract We introduce a novel kind of robustness
More informationConvex Optimization M2
Convex Optimization M2 Lecture 3 A. d Aspremont. Convex Optimization M2. 1/49 Duality A. d Aspremont. Convex Optimization M2. 2/49 DMs DM par email: dm.daspremont@gmail.com A. d Aspremont. Convex Optimization
More informationLecture Note 5: Semidefinite Programming for Stability Analysis
ECE7850: Hybrid Systems:Theory and Applications Lecture Note 5: Semidefinite Programming for Stability Analysis Wei Zhang Assistant Professor Department of Electrical and Computer Engineering Ohio State
More information2.098/6.255/ Optimization Methods Practice True/False Questions
2.098/6.255/15.093 Optimization Methods Practice True/False Questions December 11, 2009 Part I For each one of the statements below, state whether it is true or false. Include a 1-3 line supporting sentence
More informationOn the Adaptivity Gap in Two-Stage Robust Linear Optimization under Uncertain Constraints
On the Adaptivity Gap in Two-Stage Robust Linear Optimization under Uncertain Constraints Pranjal Awasthi Vineet Goyal Brian Y. Lu July 15, 2015 Abstract In this paper, we study the performance of static
More informationIncorporating Asymmetric Distributional Information. in Robust Value-at-Risk Optimization
Incorporating Asymmetric Distributional Information in Robust Value-at-Risk Optimization Karthik Natarajan Dessislava Pachamanova Melvyn Sim This version: May 2007 Forthcoming in Management Science Abstract
More information