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38
Selected topics in robust convex optimization
 MATH. PROG. B, THIS ISSUE
, 2007
"... Robust Optimization is a rapidly developing methodology for handling optimization problems affected by nonstochastic “uncertainbutbounded” data perturbations. In this paper, we overview several selected topics in this popular area, specifically, (1) recent extensions of the basic concept of robu ..."
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Cited by 35 (2 self)
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Robust Optimization is a rapidly developing methodology for handling optimization problems affected by nonstochastic “uncertainbutbounded” data perturbations. In this paper, we overview several selected topics in this popular area, specifically, (1) recent extensions of the basic concept of robust counterpart of an optimization problem with uncertain data, (2) tractability of robust counterparts, (3) links between RO and traditional chance constrained settings of problems with stochastic data, and (4) a novel generic application of the RO methodology in Robust Linear Control.
A Branch and Bound Method for Stochastic Integer Problems under Probabilistic Constraints
, 2001
"... Stochastic integer programming problems under probabilistic constraints are considered. Deterministic equivalent formulations of the original problem are obtained by using pefficient points of the probability distribution function. ..."
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Cited by 22 (0 self)
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Stochastic integer programming problems under probabilistic constraints are considered. Deterministic equivalent formulations of the original problem are obtained by using pefficient points of the probability distribution function.
An exact solution approach for portfolio optimization problems under stochastic and integer constraints
 200 18.56 4.79 3.57 17.33 7.73 0.03 15.50 9.50 2.00 600 49.60 8.33 2.22 42.32 9.73 0.03 34.46 10.39 2.00 1000 96.15 10.19 2.38 94.93 12.97 0.03 90.25 15.38 2.00 20 200 34.05 9.06 3.11 27.11 10.97 1.10 21.23 12.00 2.00 600 96.98 9.51 4.22 79.78 12.00 1.10
, 2009
"... In this paper, we study extensions of the classical Markowitz ’ meanvariance portfolio optimization model. First, we consider that the expected asset returns are stochastic by introducing a probabilistic constraint imposing that the expected return of the constructed portfolio must exceed a prescr ..."
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Cited by 21 (3 self)
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In this paper, we study extensions of the classical Markowitz ’ meanvariance portfolio optimization model. First, we consider that the expected asset returns are stochastic by introducing a probabilistic constraint imposing that the expected return of the constructed portfolio must exceed a prescribed return level with a high confidence level. We study the deterministic equivalents of these models. In particular, we define under which types of probability distributions the deterministic equivalents are secondorder cone programs, and give exact or approximate closedform formulations. Second, we account for realworld trading constraints, such as the need to diversify the investments in a number of industrial sectors, the nonprofitability of holding small positions and the constraint of buying stocks by lots, modeled with integer variables. To solve the resulting problems, we propose an exact solution approach in which the uncertainty in the estimate of the expected returns and the integer trading restrictions are simultaneously considered. The proposed algorithmic approach rests on a nonlinear branchandbound algorithm which features two new branching rules. The first one is a static rule, called idiosyncratic risk branching, while the second one is dynamic and called portfolio risk branching. The proposed branching rules are implemented and tested using the opensource framework of the solver Bonmin. The comparison of the computational results obtained with standard MINLP solvers and with the proposed approach shows the effectiveness of this latter which permits to solve to optimality problems with up to 200 assets in a reasonable amount of time.
Logarithmic concave measures and related topics
 Stochastic Programming
, 1980
"... Results obtained in the last few years are summarized in connection with logarithmic concave measures and related topics. A new and simple proof is presented concerning the basic theorem of logarithmic concave measures stating that if the probability measure P in R m is generated by a logarithmic co ..."
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Cited by 19 (0 self)
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Results obtained in the last few years are summarized in connection with logarithmic concave measures and related topics. A new and simple proof is presented concerning the basic theorem of logarithmic concave measures stating that if the probability measure P in R m is generated by a logarithmic concave density, A, B are convex subsets of R m and 0 < <1, then P ( A+(1;)B) [P (A)] [P (B)] 1;. The most important convolution theorems belonging to the subject are collected. The notion of convex measures and an important theorem concerning these are formulated. Special distributions are analysed. Properties of special constraint and objective functions which are derived from the theorems presented in the paper are described.
IIS BranchandCut for Joint ChanceConstrained Stochastic Programs and Application to Optimal Vaccine Allocation
 European Journal of Operational Research
"... We present a new method for solving stochastic programs with joint chance constraints with random technology matrices and discretely distributed random data. The problem can be reformulated as a largescale mixed 01 integer program. We derive a new class of optimality cuts called IIS cuts and apply ..."
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Cited by 13 (0 self)
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We present a new method for solving stochastic programs with joint chance constraints with random technology matrices and discretely distributed random data. The problem can be reformulated as a largescale mixed 01 integer program. We derive a new class of optimality cuts called IIS cuts and apply them to our problem. The cuts are based on irreducibly infeasible subsets (IIS) of an LP defined by requiring that all scenarios be satisfied. We propose an efficient method for improving the upper bound of the problem when no cut can be found. We derive and implement a branchandcut algorithm based on IIS cuts, and refer to this algorithm as the IIS BranchandCut algorithm. We report on computational results with several test instances from optimal vaccine allocation and a production planning problem from the literature. The computational results are very promising as the IIS branchandcut algorithm gives significantly better results than a stateoftheart commercial solver.
The Probabilistic Set Covering Problem
 OPERATIONS RESEARCH
, 2001
"... In a probabilistic set covering problem the right hand side is a random binary vector and the covering constraint has to be satisfied with some prescribed probability. We analyse the structure of the set of probabilistically efficient points of binary random vectors, develop methods for their enu ..."
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Cited by 13 (0 self)
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In a probabilistic set covering problem the right hand side is a random binary vector and the covering constraint has to be satisfied with some prescribed probability. We analyse the structure of the set of probabilistically efficient points of binary random vectors, develop methods for their enumeration and propose specialized branch and bound algorithms for probabilistic set covering problems.
An Efficient Trajectory Method for Probabilistic ProductionInventoryDistribution Problems
, 2007
"... We consider a supply chain operating in an uncertain environment: The customers’ demand is characterized by a discrete probability distribution. A probabilistic programming approach is adopted for constructing an inventoryproductiondistribution plan over a multiperiod planning horizon. The plan do ..."
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Cited by 12 (4 self)
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We consider a supply chain operating in an uncertain environment: The customers’ demand is characterized by a discrete probability distribution. A probabilistic programming approach is adopted for constructing an inventoryproductiondistribution plan over a multiperiod planning horizon. The plan does not allow the backlogging of the unsatisfied demand, and minimizes the costs of the supply chain while enabling it to reach a prescribed nonstockout service level. It is a strategic plan that hedges against undesirable outcomes, and that can be adjusted to account for possible favorable realizations of uncertain quantities. A modular, integrated, and computationally tractable method is proposed for the solution of the associated stochastic mixedinteger optimization problems containing joint probabilistic constraints with dependent righthand side variables. The concept of pefficiency is used to construct a finite number of demand trajectories, which in turn are employed to solve problems with joint probabilistic constraints. We complement this idea by designing a preordered setbased preprocessing algorithm that selects a subset of promising pefficient demand trajectories. Finally, to solve the resulting disjunctive mixedinteger programming problem, we implement a special columngeneration algorithm that limits the risk of congestion in the resources of the supply chain. The methodology is validated on an industrial problem faced by a large chemical supply chain and turns out to be very efficient: it finds a solution with a minimal integrality gap and provides substantial cost savings.
MIP reformulations of the probabilistic set covering problem
, 2007
"... In this paper we address the following probabilistic version (PSC) of the set covering problem: min{cx  P(Ax ≥ ξ) ≥ p, xj ∈ {0, 1}N} where A is a 01 matrix, ξ is a random 01 vector and p ∈ (0, 1] is the threshold probability level. We formulate (PSC) as a mixed integer nonlinear program (MINLP ..."
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Cited by 11 (1 self)
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In this paper we address the following probabilistic version (PSC) of the set covering problem: min{cx  P(Ax ≥ ξ) ≥ p, xj ∈ {0, 1}N} where A is a 01 matrix, ξ is a random 01 vector and p ∈ (0, 1] is the threshold probability level. We formulate (PSC) as a mixed integer nonlinear program (MINLP) and linearize the resulting (MINLP) to obtain a MIP reformulation. We introduce the concepts of pinefficiency and polarity cuts. While the former is aimed at reducing the number of constraints in our model, the later is used as a strengthening device to obtain stronger formulations. A hierarchy of relaxations for (PSC) is introduced, and fundamental relationships between the relaxations are established culminating with a MIP reformulation of (PSC) with no additional integer constrained variables. Simplifications of the MIP model which result when one of the following conditions hold are briefly discussed: A is a balanced matrix, A has the circular ones property, the components of ξ are pairwise independent, the distribution function of ξ is a stationary distribution or has the socalled disjunctive shattering property. We corroborate our theoretical findings by an extensive computational experiment on a testbed consisting of almost 10,000 probabilistic instances. This testbed was created using deterministic instances from the literature and consists of probabilistic variants of the setcovering model and capacitated versions of facility location, warehouse location and kmedian models. Our computational results show that our procedure is orders of magnitude faster than any of the existing approaches to solve (PSC), and in many cases can reduce hours of computing time to fraction of seconds.
A BranchandCut Decomposition Algorithm for Solving ChanceConstrained Mathematical Programs with Finite Support
, 2013
"... We present a new approach for exactly solving chanceconstrained mathematical programs having discrete distributions with finite support and random polyhedral constraints. Such problems have been notoriously difficult to solve due to nonconvexity of the feasible region, and most available methods ar ..."
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Cited by 9 (2 self)
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We present a new approach for exactly solving chanceconstrained mathematical programs having discrete distributions with finite support and random polyhedral constraints. Such problems have been notoriously difficult to solve due to nonconvexity of the feasible region, and most available methods are only able to find provably good solutions in certain very special cases. Our approach uses both decomposition, to enable processing subproblems corresponding to one possible outcome at a time, and integer programming techniques, to combine the results of these subproblems to yield strong valid inequalities. Computational results on a chanceconstrained formulation of a resource planning problem inspired by a call center staffing application indicate the approach works significantly better than both an existing mixedinteger programming formulation and a simple decomposition approach that does not use strong valid inequalities. We also demonstrate how the approach can be used to efficiently solve for a sequence of risk levels, as would be done when solving for the efficient frontier of risk and cost.
Finite adaptability in multistage linear optimization
, 2007
"... In multistage problems, decisions are implemented sequentially, and thus may depend on past realizations of the uncertainty. Examples of such problems abound in applications of stochastic control and operations research; yet, where robust optimization has made great progress in providing a tractable ..."
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Cited by 8 (2 self)
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In multistage problems, decisions are implemented sequentially, and thus may depend on past realizations of the uncertainty. Examples of such problems abound in applications of stochastic control and operations research; yet, where robust optimization has made great progress in providing a tractable formulation for a broad class of singlestage optimization problems with uncertainty, multistage problems present significant tractability challenges. In this paper we consider an adaptability model designed with problems with discrete second stage variables in mind. We propose a hierarchy of increasing adaptability that bridges the gap between the static robust formulation, and the fully adaptable formulation. We study the geometry, complexity, formulations, algorithms, examples and computational results for finite adaptability. In contrast to the model of affine adaptability proposed in [2], our proposed framework can accommodate discrete variables. In terms of performance for continuous linear optimization, the two frameworks are complementary, in the sense that we provide examples that the proposed framework provides stronger solutions and vice versa. We prove a positive tractability result in the regime where we expect finite adaptability to perform well, and illustrate this claim and the finite adaptability methodology, with an application from Air Traffic Control. 1