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We consider the problem of optimal decision making under uncertainty but assume that the decision maker's utility function is not completely known. Instead, we consider all the utilities that meet some criteria, such as preferring certain lotteries over other lotteries and being risk averse, S-shaped, or prudent. These criteria extend the ones used in the first- and second-order stochastic dominance framework. We then give tractable formulations for such decision-making problems. We formulate them as robust utility maximization problems, as optimization problems with stochastic dominance constraints, and as robust certainty equivalent maximization problems. We use a portfolio allocation problem to illustrate our results. Keywords: expected utility; robust optimization; stochastic dominance; certainty equivalent History: Received June 29, 2012; accepted August 5, 2014, by Dimitris Bertsimas, optimization., 1. Introduction This paper questions a key and rarely challenged assumption of decision making under uncertainty: that decision makers can always, after a tolerable amount of introspective questioning, clearly identify [...]

Conditional value at risk (CVaR) is widely used to account for the preferences of a risk-averse agent in extreme loss scenarios. To study the effectiveness of randomization in interdiction problems with an interdictor that is both risk- and ambiguity-averse, we introduce a distributionally robust maximum flow network interdiction problem in which the interdictor randomizes over the feasible interdiction plans in order to minimize the worst case CVaR of the maximum flow with respect to both the unknown distribution of the capacity of the arcs and the interdictor's own randomized strategy. Using the size of the uncertainty set, we control the degree of conservatism in the model and reformulate the interdictor's distributionally robust optimization problem as a bilinear optimization problem. For solving this problem to any given optimality level, we devise a spatial branch-and-bound algorithm that uses the McCormick inequalities and reduced reformulation linearization technique to obtain a convex relaxation of the problem. We also develop a column-generation algorithm to identify the optimal support of the convex relaxation, which is then used in the coordinate descent algorithm to determine the upper bounds. The efficiency and convergence of the spatial branch-and-bound algorithm is established in numerical experiments. Further, our numerical experiments show that randomized strategies can have significantly better performance than optimal deterministic ones. History: Accepted by David Alderson, Area Editor for Network Optimization: Algorithms & Applications. Funding: The first author's research is supported by a Group for Research in Decision Analysis postdoctoral fellowship and Fonds de Recherche du Québec–Nature et Technologies postdoctoral research scholarship [Grants 275296 and 301065]. The second author acknowledges support from the Canadian Natural Sciences and Engineering Research Council and the Canada Research Chair program [Grants RGPIN-2016-05208, 492997-2016, and 950-230057]. Supplemental Material: The e-companion is available at https://doi.org/10.1287/ijoc.2022.1257. The data file and codes are posted on GitHub (https://github.com/Utsav19/Value-of-Randomization). [ABSTRACT FROM AUTHOR]

Robust optimization is a powerful means to handle optimization problems where there is a set of parameters that are uncertain. The effectiveness of the method is especially noticeable when these parameters are only known to lie inside some uncertainty region. Unfortunately, there are important computational considerations that have prevented the methodology from being fully adopted in fields of practice where the cost function that needs to be robustified is nonlinear with respect to such parameters. In this paper, we propose a new robust optimization formulation that circumvent the computational burden in problems where the cost decomposes as the sum of costs that each involve a single decision variable. This is done by exploiting the fact that in this formulation the worst-case cost function can be expressed as a convex combination between a nominal and an upper-bounding cost function. One can still control the conservatism of the robust solution by adjusting how many terms of this total cost can simultaneously reach their respective most pessimistic value. To demonstrate the potential of this "practicable robust counterpart" formulation, we present how it can be employed for the robust optimization of packet routing on a telecommunication network with congestion. In such problems, an important source of uncertainty stems from the queueing delay, which is usually approximated by a nonlinear convex function using the theory of M/M/1 queues. Computational results on a large number of problem instances of realistic size confirmthat it is possible to identify robust solutions that significantly outperform a deterministic approach in terms of both the amount of congestion and the risks of excessive congestion. Moreover, our proposed method also improves significantly the quality of solutions that are obtained using two other natural approximation schemes. [ABSTRACT FROM AUTHOR]

Robust optimization is a methodology that has gained a lot of attention in the recent years. This is mainly due to the simplicity of the modeling process and ease of resolution even for large scale models. Unfortunately, the second property is usually lost when the cost function that needs to be "robustified" is not concave (or linear) with respect to the perturbing parameters. In this paper we study robust optimization of sums of piecewise linear functions over polyhedral uncertainty set. Given that these problems are known to be intractable, we propose a new scheme for constructing conservative approximations based on the relaxation of an embedded mixed-integer linear program and relate this scheme to methods that are based on exploiting affine decision rules. Our new scheme gives rise to two tractable models that, respectively, take the shape of a linear program and a semidefinite program, with the latter having the potential to provide solutions of better quality than the former at the price of heavier computations. We present conditions under which our approximation models are exact. In particular, we are able to propose the first exact reformulations for a robust (and distributionally robust) multi-item newsvendor problem with budgeted uncertainty set and a reformulation for robust multiperiod inventory problems that is exact whether the uncertainty region reduces to a L1-norm ball or to a box. An extensive set of empirical results will illustrate the quality of the approximate solutions that are obtained using these two models on randomly generated instances of the latter problem. [ABSTRACT FROM AUTHOR]

This paper considers a distributionally robust version of a quadratic knapsack problem. In this model, a subsets of items is selected to maximizes the total profit while requiring that a set of knapsack constraints be satisfied with high probability. In contrast to the stochastic programming version of this problem, we assume that only part of the information on random data is known, i.e., the first and second moment of the random variables, their joint support, and possibly an independence assumption. As for the binary constraints, special interest is given to the corresponding semidefinite programming (SDP) relaxation. While in the case that the model only has a single knapsack constraint we present an SDP reformulation for this relaxation, the case of multiple knapsack constraints is more challenging. Instead, two tractable methods are presented for providing upper and lower bounds (with its associated conservative solution) on the SDP relaxation. An extensive computational study is given to illustrate the tightness of these bounds and the value of the proposed distributionally robust approach. [ABSTRACT FROM AUTHOR]

Recently, coinciding with and perhaps driving the increased popularity of prediction markets, several novel pari-mutuel mechanisms have been developed such as the logarithmic market-scoring rule (LMSR), the cost-function formulation of market makers, utility-based markets, and the sequential convex pari-mutuel mechanism (SCPM). In this work, we present a convex optimization framework that unifies these seemingly unrelated models for centrally organizing contingent claims markets. The existing mechanisms can be expressed in our unified framework by varying the choice of a concave value function. We show that this framework is equivalent to a convex risk minimization model for the market maker. This facilitates a better understanding of the risk attitudes adopted by various mechanisms. The unified framework also leads to easy implementation because we can now find the cost function of a market maker in polynomial time by solving a simple convex optimization problem. In addition to unifying and explaining the existing mechanisms, we use the generalized framework to derive necessary and sufficient conditions for many desirable properties of a prediction market mechanism such as proper scoring, truthful bidding (in a myopic sense), efficient computation, controllable risk measure, and guarantees on the worst-case loss. As a result, we develop the first proper, truthful, risk-controlled, loss-bounded (independent of the number of states) mechanism; none of the previously proposed mechanisms possessed all these properties simultaneously. Thus, our work provides an effective tool for designing new prediction market mechanisms. We also discuss possible applications of our framework to dynamic resource pricing and allocation in general trading markets. [ABSTRACT FROM AUTHOR]

In this paper, we study how uncertainties weighing on the climate system impact the optimal technological pathways the world energy system should take to comply with stringent mitigation objectives. We use the TIAM-World model that relies on the TIMES modelling approach. Its climate module is inspired by the DICE model. Using robust optimization techniques, we assess the impact of the climate system parameter uncertainty on energy transition pathways under various climate constraints. Unlike other studies we consider all the climate system parameters which is of primary importance since: (i) parameters and outcomes of climate models are all inherently uncertain (parametric uncertainty); and (ii) the simplified models at stake summarize phenomena that are by nature complex and non-linear in a few, sometimes linear, equations so that structural uncertainty is also a major issue. The use of robust optimization allows us to identify economic energy transition pathways under climate constraints for which the outcome scenarios remain relevant for any realization of the climate parameters. In this sense, transition pathways are made robust. We find that the abatement strategies are quite different between the two temperature targets. The most stringent one is reached by investing massively in carbon removal technologies such as bioenergy with carbon capture and storage (BECCS) which have yields much lower than traditional fossil fuelled technologies. [ABSTRACT FROM AUTHOR]