Robust Optimization for Decision-making under Endogenous Uncertainty Nikolaos H. Lappas, Chrysanthos E. Gounaris* Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, PA, 15213 Abstract This paper contemplates the use of robust optimization as a framework for addressing prob- lems that involve endogenous uncertainty, i.e., uncertainty that is affected by the decision maker's strategy. To that end, we extend generic polyhedral uncertainty sets typically con- sidered in robust optimization into sets that depend on the actual decisions. We present the derivation of robust counterpart models in this setting, and we discuss relevant algorithmic con- siderations for solving these models to guaranteed optimality. Besides capturing the functional changes in parameter correlations that may be induced by given decisions, we show how the use of our decision-dependent uncertainty sets allows us to also eradicate conservatism effects from parameters that become irrelevant in view of the optimal decisions. We quantify these benefits via a number of case studies, demonstrating our proposed framework's versatility to be utilized in the context of various applications. Keywords: Robust Optimization, Endogenous Uncertainty, Decision-dependent Uncer- tainty Sets 1 Introduction Uncertainty is inherent in virtually every system we wish to optimize. Parameters affected by various external forces such as market prices and demand, unexpected disruptive events such as equipment malfunctions and natural disasters, or simply incomplete information about the system under study may render solutions of deterministic optimization models suboptimal or even infeasible 1 Author to whom all correspondence should be addressed ([email protected]). 1 when parameter realizations deviate from their nominal values. To that end, multiple approaches have been proposed so as to account for uncertainty during the decision-making process. The various alternative methodologies that have been developed serve different purposes, and selecting which one to adopt should be based on careful examination of the characteristics of the application in question. For example, when the number of uncertain parameters is relatively small, multi-parametric programming [1] can map the parameter space in order to offer closed- form solutions of optimization problems in terms of the former and to provide robustness estimates of any given solution. When detailed probabilistic information about the system parameters is available, often in the form of discrete scenario trees, stochastic programming [2] can be utilized to optimize system performance in expectation. Bounds on the variability of such performance can then be applied as explicit constraints on various statistically meaningful metrics [3]. In stochastic programming, the assumption of full recourse is commonly utilized and is often implemented by penalizing infeasibilities in the objective function. It should be mentioned, however, that this practice would not be suitable in applications where constraint violations cannot be tolerated (e.g., due to system safety concerns), are not meaningful (e.g., equipment physical limitations), or cannot be fairly \monetized." On the other hand, robust optimization (RO) [4{6] offers an attractive option for applications where distributional information about the uncertainty is limited and/or where solution feasibility is top priority. For such settings, RO seeks solutions that remain feasible for any possible uncertainty realization from within a postulated uncertainty set, which captures applicable correlations among uncertain parameters. Although one can derive such sets based on probabilistic information [7, 8], precise knowledge of probability distributions is not typically required to construct such sets. Multiple types of uncertainty sets (e.g., polyhedral, ellipsoidal, cardinality-constrained budgets) can be used in this context, exploiting correlations among uncertain parameters as a mechanism to control the trade-off between robustness and performance. A common characteristic across traditional uncertainty sets utilized in RO is their constant nature; that is, the range of parameter realizations they admit does not depend on the values we choose for the decision variables. This fact leads to an important limitation, namely that constant sets do not suffice to model settings where one's decisions directly affect the underlying probability distribution from which parameter realizations draw (e.g., entering a market sooner provides access 2 to a larger demand). Furthermore, a subtle manifestation arises in cases where one's decisions render a model parameter that is referenced in the uncertainty set physically meaningless (e.g., the yield of a process step that was not selected in the optimal flowsheet). Not only would it be unrealistic to postulate a correlation involving such a parameter, doing so might lead to overly conservative solutions as a result of an effective relaxation of the uncertainty set's projection to the space of parameters that remain relevant in view of the optimal decisions. In order to overcome the above challenges, we recently proposed the use of decision-dependent uncertainty sets (DDUS) in the context of robust optimization models for process scheduling appli- cations [9]. In that work, the effects of task processing times were removed from the uncertainty set whenever the decision was taken to not execute such tasks, offering a more realistic representation of the uncertainty encountered in the actual application, as well as leading to less conservative solutions compared to robust solutions reported in prior literature. Expanding upon a previously published shorter version [10], the contributions of the current paper are four-fold: • We classify decision-making contexts involving uncertain parameters of endogenous nature for which the use of DDUS is warranted. • We propose DDUS of generic polyhedral form that feature decision-dependency in both left- and right-hand sides, and we discuss ways to instantiate such sets. • We derive the robust counterpart model in the context of our proposed DDUS, and we discuss applicable algorithmic approaches to solve the former to guaranteed optimality. • We assess the performance benefits as well as the computational burden associated with using our proposed DDUS over a series of case studies that are addressed with RO for the first time in the open literature. The remainder of this paper is structured as follows. We begin by providing a brief overview of the static RO methodology in Section 2. In Section 3, we discuss the possibility for uncertain parameters to exhibit an endogenous nature, and we introduce our novel DDUS that allow us to model this setting. We then derive the robust counterpart under DDUS in Section 4, while in Section 5, we present a comprehensive computational study involving a number of case studies 3 stemming from a diverse set of application contexts. We conclude the paper with some final remarks in Section 6. Notation. Lowercase letters with regular typeface denote scalars (e.g., a), lowercase letters with bold typeface denote column vectors (e.g., a), while uppercase letters in bold typeface denote th matrices (e.g., A). We use jaj to indicate the size of vector a, and we use ai to denote its i element. We also use e to denote the vector of ones, 0 to denote the vector of zeros, E to denote the matrix of all ones, and O to denote the matrix of all zeros. Sizes for these are to be inferred from the expressions in which they participate. The ◦ operator corresponds to the element-wise product of two equally-sized vectors or matrices, while for a quantity a, we use a and a to represent this quantity's applicable lower and upper bound, respectively. Finally, an equation involving vector or matrix terms (e.g., a ≤ b) is to be viewed as a set of equations referencing the terms element-wise (e.g., ai ≤ bi 8 i), while an implication or equivalence with a set of equations in its sides (e.g., fa = bg , fc ≤ dg) is also to be viewed element-wise (e.g., fai = big , fci ≤ dig 8 i). 2 Static Robust Optimization Let us consider the mixed-integer linear optimization problem (1),1 which features continuous decision variables x, discrete decision variables w, and inequality constraints indexed over the set M. min x1 x;w jxj jwj s.t. x 2 R ; w 2 f0; 1g (1) > > > > > cmx + q Cmx + rmw + q Rmw ≤ am + q bm 8 m 2 M (x; w) 2 F For ease of exposition, the various expressions in the constraints have been partitioned into terms that do and terms that do not involve parameters q, which we shall consider to be uncertain jqj and about to realize from within an uncertainty set Q ⊆ R . The quantities cm, Cm, rm, Rm, am, and bm are therefore constants known with certainty, for all m 2 M, while the set jxj jwj F ⊆ R × f0; 1g is introduced to represent the space that results from intersecting all linear constraints (both equalities and inequalities) that do not reference any of the uncertain parameters 1 We remark that, although in this paper we focus on problems that can be represented via mixed-integer linear formulations, the RO methodology can be extended to nonlinear models as well. 4 q. For clarity, we mention that we consider all constraints in the set M to reference at least one parameter from q; that is, at least one of the elements of Cm, Rm, or bm must be non-zero, for each constraint m 2 M. Furthermore, the objective function has been brought into the set of constraints using a standard epigraph reformulation (where, without loss of generality, the continuous variable x1 is used to denote the epigraph variable). Consequently, any mixed-integer linear problem that does not reference uncertain parameters in equality constraints2 can be equivalently transformed to the form (1). We also highlight that this formulation can accommodate uncertain parameters in both left- and right-hand sides as well as in objective function coefficients. In this setting, all decisions are to be taken in a single stage, before the actual realizations of the uncertain parameters are observed.