In this paper, an algorithm for maximizing a convex function over a convex feasible set is proposed. The algorithm, called CoMax, consists of two phases: in phase 1, a feasible starting point is obtained that is used in a gradient ascent algorithm in phase 2. The main contribution of the paper is connected to phase 1; five different methods are used to approximate the original NP-hard problem of maximizing a convex function (MCF) by a tractable convex optimization problem. All the methods use the minimizer of the convex objective function in their construction. In phase 2, the gradient ascent algorithm yields stationary points to the MCF problem. The performance of CoMax is tested on a wide variety of MCF problems, demonstrating its efficiency. History: Accepted by Antonio Frangioni, Area Editor for Design & Analysis of Algorithms–Continuous. Funding: This work was supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek [Grant 406.17.511]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplementary Information [ https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.1238 ] or is available from the IJOC GitHub software repository ( https://github.com/INFORMSJoC ) at [ http://dx.doi.org/10.5281/zenodo.6884872 ].

In this paper we consider uncertain second-order cone (SOC) and semidefinite programming (SDP) constraints with polyhedral uncertainty, which are in general computationally intractable. We propose to reformulate an uncertain SOC or SDP constraint as a set of adjustable robust linear optimization constraints withan ellipsoidal or semidefinite representable uncertainty set, respectively. The resulting adjustable problem can then (approximately) be solved by using adjustable robust linear optimization techniques. For example, we show that if linear decision rules are used, then the final robust counterpart consists of SOC or SDP constraints, respectively, which have the same computational complexity as the nominal version of the original constraints. We propose an efficient method to obtain good lower bounds. Moreover, we extend our approach to other classes of robust optimization problems, such as nonlinear problems that contain waitand-see variables or linear problems that contain bilinear uncertainty. Numerically, we apply our approach to reformulate the problem on finding the minimum volume circumscribing ellipsoid of a polytope, and solvethe resulting reformulation with linear and quadratic decision rules as well as Fourier-Motzkin elimination. We demonstrate the effectiveness and efficiency of the proposed approach by comparing it with the state-ofthe-art copositive approach. Moreover, we apply the proposed approach to a robust regression problem and a robust sensor network problem, and use linear decision rules to solve the resulting adjustable robust linear optimization problems, which solves the problem to (near) optimality.

Although robust optimization is a powerful technique in dealing with uncertainty in optimization, its solutions can be too conservative. More specifically, it can lead to an objective value much worse than the nominal solution or even to infeasibility of the robust problem. In practice, this can lead to robust solutions being disregarded in favor of the nominal solution. This conservatism is caused by both the constraint-wise approach of robust optimization and its core assumption that all constraints are hard for all scenarios in the uncertainty set. This paper seeks to alleviate this conservatism by proposing an alternative robust formulation that condenses all uncertainty into a single constraint, binding the worst-case expected violation in the original constraints from above. Using recent results in distributionally robust optimization, the proposed formulation is shown to be tractable for both right- and left-hand side uncertainty. A computational study is performed with problems from the NETLIB library. For some problems, the percentage of uncertainty is magnified fourfold in terms of increase in objective value of the standard robust solution compared with the nominal solution, whereas we find solutions that safeguard against over half the violation at only a 10th of the cost in objective value. For problems with an infeasible standard robust counterpart, the suggested approach is still applicable and finds both solutions that safeguard against most of the uncertainty at a low price in terms of objective value.