Radius of robust feasibility formulas for classes of convex programs with uncertain polynomial constraints (original) (raw)
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Journal of Optimization Theory and Applications, 2021
The radius of robust feasibility provides a numerical value for the largest possible uncertainty set that guarantees robust feasibility of an uncertain linear conic program. This determines when the robust feasible set is non-empty. Otherwise the robust counterpart of an uncertain program is not well-defined as a robust optimization problem. In this paper, we address a key fundamental question of robust optimization: How to compute the radius of robust feasibility of uncertain linear conic programs, including linear programs? We first provide computable lower and upper bounds for the radius of robust feasibility for general uncertain linear conic programs under the commonly used ball uncertainty set. We then provide important classes of linear conic programs where the bounds are calculated by finding the optimal values of related semidefinite linear programs, among them uncertain semidefinite programs, uncertain second-order cone programs and uncertain support vector machine problems. In the case of an uncertain linear program, the exact formula allows us to calculate the radius by finding the optimal value of an associated second-order cone program.
Robust convex quadratically constrained programs
Mathematical Programming, 2003
In this paper we study robust convex quadratically constrained programs, a subset of the class of robust convex programs introduced by Ben-Tal and Nemirovski [4]. Unlike [4], our focus in this paper is to identify uncertainty structures that allow the corresponding robust quadratically constrained programs to be reformulated as second-order cone programs. We propose three classes of uncertainty sets that satisfy this criterion and present examples where these classes of uncertainty sets are natural.
Robust solutions of uncertain linear programs
Operations Research Letters, 1999
We treat in this paper Linear Programming (LP) problems with uncertain data. The focus is on uncertainty associated with hard constraints: those which must be satisfied, whatever is the actual realization of the data (within a prescribed uncertainty set). We suggest a modeling methodology whereas an uncertain LP is replaced by its Robust Counterpart (RC). We then develop the analytical and computational optimization tools to obtain robust solutions of an uncertain LP problem via solving the corresponding explicitly stated convex RC program. In particular, it is shown that the RC of an LP with ellipsoidal uncertainty set is computationally tractable, since it leads to a conic quadratic program, which can be solved in polynomial time. If the vector c is also uncertain, we could look at the equivalent formulation of (1): min x,t t : c T x ≤ t, Ax ≥ b and thus without loss of generality we may restrict the uncertainty to the constraints only.
Gulf journal of mathematics, 2024
In this work, we study an uncertain quadratic optimization problem by using the concept of stability radius. That allows to prove robustness of uncertain quadratic optimization models. Firstly, we determine the stability radius of this problem by applying Ascoli formula. Secondly, we show that its robust counterpart has at least an optimal solution. Finally, we deal with an application and provide some numerical methods for such an uncertain quadratic optimization problem.
Adjustable robust solutions of uncertain linear programs
Mathematical Programming, 2004
We treat in this paper Linear Programming (LP) problems with uncertain data. The focus is on uncertainty associated with hard constraints: those which must be satisfied, whatever is the actual realization of the data (within a prescribed uncertainty set). We suggest a modeling methodology whereas an uncertain LP is replaced by its Robust Counterpart (RC). We then develop the analytical and computational optimization tools to obtain robust solutions of an uncertain LP problem via solving the corresponding explicitly stated convex RC program. In particular, it is shown that the RC of an LP with ellipsoidal uncertainty set is computationally tractable, since it leads to a conic quadratic program, which can be solved in polynomial time. If the vector c is also uncertain, we could look at the equivalent formulation of (1): min x,t t : c T x ≤ t, Ax ≥ b and thus without loss of generality we may restrict the uncertainty to the constraints only.
European Journal of Operational Research, 2022
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Operations Research, 2014
We propose a new way to derive tractable robust counterparts of a linear program based on the duality between the robust (“pessimistic”) primal problem and its “optimistic” dual. First we obtain a new convex reformulation of the dual problem of a robust linear program, and then show how to construct the primal robust solution from the dual optimal solution. Our result allows many new uncertainty regions to be considered. We give examples of tractable uncertainty regions that were previously intractable. The results are illustrated by solving a multi-item newsvendor problem. We also apply the new method to the globalized robust counterpart scheme and show its tractability.
Uncertain convex programs: randomized solutions and confidence levels
Mathematical Programming, 2005
Many engineering problems can be cast as optimization problems subject to convex constraints that are parameterized by an uncertainty or 'instance' parameter. A recently emerged successful paradigm for attacking these problems is robust optimization, where one seeks a solution which simultaneously satisfies all possible constraint instances. In practice, however, the robust approach is effective only for problem families with rather simple dependence on the instance parameter (such as affine or polynomial), and leads in general to conservative answers, since the solution is usually computed by transforming the original semi-infinite problem into a standard one, by means of relaxation techniques.
Robust Solutions of MultiObjective Linear Semi-Infinite Programs under Constraint Data Uncertainty
Siam Journal on Optimization, 2014
The multiobjective optimization model studied in this paper deals with simultaneous minimization of finitely many linear functions subject to an arbitrary number of uncertain linear constraints. We first provide a radius of robust feasibility guaranteeing the feasibility of the robust counterpart under affine data parametrization. We then establish dual characterizations of robust solutions of our model that are immunized against data uncertainty by way of characterizing corresponding solutions of robust counterpart of the model. Consequently, we present robust duality theorems relating the value of the robust model with the corresponding value of its dual problem.
On Approximate Robust Counterparts of Uncertain Semidefinite and Conic Quadratic Programs
System Modeling and Optimization XX, 2003
We present efficiently verifiable sufficient conditions for the validity of specific NP-hard semi-infinite systems of semidefinite and conic quadratic constraints arising in the framework of Robust Convex Programming and demonstrate that these conditions are "tight" up to an absolute constant factor. We discuss applications in Control on the construction of a quadratic Lyapunov function for linear dynamic system under interval uncertainty.