Melvyn Sim | Searcharxiv

arXiv2025

Modified Polyhedral Method for Elicitation of Shape-Free Utility and Conservatism Reduction in Robust Optimization

In this paper, we propose a modified polyhedral method to elicit a decision maker's (DM's) nonlinear univariate utility function, which does not rely on explicit information about the shape structure, Lipschitz modulus, and the inflection point of the utility. The method is inspired by Toubia et al. (2004) for elicitation of the linear multi-variate utility and the success of the modification needs to overcome two main difficulties. First, we use the continuous piecewise linear function (PLF) to approximate the nonlinear utility and represent the PLF in terms of the vector of increments of linear pieces. Subsequently, elicitation of the nonlinear utility corresponds to reducing the polyhedral feasible set of the vectors of increments. Second, we reduce the size of the polyhedron by successive hyperplane cuts constructed by adaptively generating new queries (pairwise comparison lotteries) where the parameters of the lotteries are obtained by solving some optimization problems. In this reduction procedure, direction error of the cut hyperplane may occur due to the PLF approximation error. To tackle the issue, we develop a strategy by adding the support points of new lotteries to the set of breakpoints of the PLF. As an application, we use all the responses to the queries to construct an ambiguity set of utility functions which allows one to make decisions based on the worst-case utility and apply the modified polyhedral method in a preference robust optimization problem with proper conservatism reduction scheme. The preliminary numerical test results show that the proposed methods work very well.

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arXiv2021

Robust Conic Satisficing

In practical optimization problems, we typically model uncertainty as a random variable though its true probability distribution is unobservable to the decision maker. Historical data provides some information of this distribution that we can use to approximately quantify the risk that depends on both the decision and the uncertainty. This empirical optimization approach is vulnerable to the issues of overfitting, which could be overcome by several data-driven robust optimization techniques. To tackle overfitting, Long et.al.(2022) propose a robust satisficing model, which is specified by a performance target and a penalty function that measures the deviation of the uncertainty from its nominal value, and yields solutions with superior out-of-sample performance. We generalize the robust satisficing framework to conic optimization problems with recourse, which has broad applications in predictive and prescriptive analytics. We derive an exact semidefinite optimization formulation for a biconvex quadratic evaluation function, with quadratic penalty and ellipsoidal support set. More importantly, under complete and bounded recourse, and a reasonably chosen polyhedral support set and penalty function, we propose safe approximations that are feasible for any reasonably chosen target. We then demonstrate that the assumption of complete and bounded recourse is not unimpeachable, and then introduce a novel perspective casting technique to derive an equivalent conic optimization problem satisfying the stated assumptions. Computationally, we showcase a study on data-driven portfolio optimization and demonstrate that the robust satisficing solutions can provide significant improvements over the solutions obtained by stochastic optimization models, including the celebrated Markowitz model, which is prone to overfitting.

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