Calculating Radius of Robust Feasibility of Uncertain Linear Conic Programs via Semi-definite Programs

We present an exact formula for the radius of robust feasibility of uncertain linear programs with a compact and convex uncertainty set. The radius of robust feasibility provides a value for the maximal `size' of an uncertainty set under which robust feasibility of the uncertain linear program can be guaranteed. By considering spectrahedral uncertainty sets, we obtain numerically tractable radius formulas for commonly used uncertainty sets of robust optimization, such as ellipsoids, balls, polytopes and boxes. In these cases, we show that the radius of robust feasibility can be found by solving a linearly constrained convex quadratic program or a minimax linear program. The results are illustrated by calculating the radius of robust feasibility of uncertain linear programs for several different uncertainty sets.

Second-order cone programming (SOCP) problems are convex optimization problems in which a linear function is minimized over the intersection of an affine linear manifold with the Cartesian product of second-order (Lorentz) cones. Linear programs, convex quadratic programs and quadratically constrained convex quadratic programs can all be formulated as SOCP problems, as can many other problems that do not fall into these three categories. These latter problems model applications from a broad range of fields from engineering, control and finance to robust optimization and combinatorial optimization. On the other hand semidefinite programming (SDP)—that is the optimization problem over the intersection of an affine set and the cone of positive semidefinite matrices—includes SOCP as a special case. Therefore, SOCP falls between linear (LP) and quadratic (QP) programming and SDP. Like LP, QP and SDP problems, SOCP problems can be solved in polynomial time by interior point methods. The computational effort per iteration required by these methods to solve SOCP problems is greater than that required to solve LP and QP problems but less than that required to solve SDP’s of similar size and structure. Because the set of feasible solutions for an SOCP problem is not polyhedral as it is for LP and QP problems, it is not readily apparent how to develop a simplex or simplex-like method for SOCP. While SOCP problems can be solved as SDP problems, doing so is not advisable both on numerical grounds and computational complexity concerns. For instance, many of the problems presented in the survey paper of Vandenberghe and Boyd [VB96] as examples of SDPs can in fact be formulated as SOCPs and should be solved as such. In §2, 3 below we give SOCP formulations for four of these examples: the convex quadratically constrained quadratic programming (QCQP) problem, problems involving fractional quadratic functions ∗RUTCOR, Rutgers University, e-mail:alizadeh@rutcor.rutgers.edu. Research supported in part by the U.S. National Science Foundation grant CCR-9901991 †IEOR, Columbia University, e-mail: gold@ieor.columbia.edu. Research supported in part by the Department of Energy grant DE-FG02-92ER25126, National Science Foundation grants DMS-94-14438, CDA-97-26385 and DMS-01-04282.

This paper presents exact dual semi-definite programs (SDPs) for robust SOS-convex polynomial optimization problems with affinely adjustable variables in the sense that the optimal values of the robust problem and its associated dual SDP are equal with the solution attainment of the dual problem. This class of robust convex optimization problems includes the corresponding quadratically constrained convex quadratic optimization problems and separable convex polynomial optimization problems, and it employs a general bounded spectrahedron uncertainty set that covers the most commonly used uncertainty sets of numerically solvable robust optimization models, such as boxes, balls and ellipsoids. As special cases, it also demonstrates that explicit exact dual SDP and second-order cone programming (SOCP) in terms of original data hold for the robust two-stage convex quadratic programs with quadratic constraints and the robust two-stage separable convex quadratic programs under an ellipsoidal uncertainty set, respectively. Finally, the paper illustrates the results via numerical implementations of the developed SDP duality scheme on adjustable robust lot-sizing problems with nonlinear costs under demand uncertainty.

Robust global error bounds for uncertain linear inequality systems with applications

In a real situation, optimization problems often involve uncertain parameters. Robust optimization is one of distribution-free methodologies based on worst-case analyses for handling such problems. In this paper, we first focus on a special class of uncertain linear programs (LPs). Applying the duality theory for nonconvex quadratic programs (QPs), we reformulate the robust counterpart as a semidefinite program (SDP) and show the equivalence property under mild assumptions. We also apply the same technique to the uncertain second-order cone programs (SOCPs) with "single" (not side-wise) ellipsoidal uncertainty. Then we derive similar results on the reformulation and the equivalence property. In the numerical experiments, we solve some test problems to demonstrate the efficiency of our reformulation approach. Especially, we compare our approach with another recent method based on Hildebrand's Lorentz positivity.

This paper addresses the issue of which strong duality holds between parametric robust semi-definite linear optimization problems and their dual programs. In the case of a spectral norm uncertainty set, it yields a corresponding strong duality result with a semi-definite programming as its dual. We also show that the dual can be reformulated as a second-order cone programming problem or a linear programming problem when the constraint uncertainty sets of parametric robust semi-definite linear programs are given in terms of affinely parameterized diagonal matrix.

Applications of second-order cone programming

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 with an 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 wait-and-see variables, linear problems that contain bilinear uncertainty, and general conic constraints. Numerically, we apply our approach to reformulate the problem on finding the minimum volume circumscribing ellipsoid of a polytope and solve the 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-of-the-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 solve the problem to (near) optimality. Summary of Contribution: Computing robust solutions for nonlinear optimization problems with uncertain second-order cone and semidefinite programming constraints are of much interest in real-life applications, yet they are in general computationally intractable. This paper proposes a computationally tractable approximation for such problems. Extensive computational experiments on (i) computing the minimum volume circumscribing ellipsoid of a polytope, (ii) robust regressions, and (iii) robust sensor networks are conducted to demonstrate the effectiveness and efficiency of the proposed approach.

Uncertain linear programming with cloud set constraints integrating fuzziness and randomness

For a mixed-integer linear problem (MIP) with uncertain constraints, the radius of robust feasibility (RRF) determines a value for the maximal size of the uncertainty set such that robust feasibility of the MIP can be guaranteed. The approaches for the RRF in the literature are restricted to continuous optimization problems. We first analyze relations between the RRF of a MIP and its continuous linear (LP) relaxation. In particular, we derive conditions under which a MIP and its LP relaxation have the same RRF. Afterward, we extend the notion of the RRF such that it can be applied to a large variety of optimization problems and uncertainty sets. In contrast to the setting commonly used in the literature, we consider for every constraint a potentially different uncertainty set that is not necessarily full-dimensional. Thus, we generalize the RRF to MIPs and to include safe variables and constraints; that is, where uncertainties do not affect certain variables or constraints. In the extended setting, we again analyze relations between the RRF for a MIP and its LP relaxation. Afterward, we present methods for computing the RRF of LPs and of MIPs with safe variables and constraints. Finally, we show that the new methodologies can be successfully applied to the instances in the MIPLIB 2017 for computing the RRF. Summary of Contribution: Robust optimization is an important field of operations research due to its capability of protecting optimization problems from data uncertainties that are usually defined via so-called uncertainty sets. Intensive research has been conducted in developing algorithmically tractable reformulations of the usually semi-infinite robust optimization problems. However, in applications it also important to construct appropriate uncertainty sets (i.e., prohibiting too conservative, intractable, or even infeasible robust optimization problems due to the choice of the uncertainty set). In doing so, it is useful to know the maximal “size” of a given uncertainty set such that a robust feasible solution still exists. In this paper, we study one notion of “size”: the radius of robust feasibility (RRF). We contribute on the theoretical side by generalizing the RRF to MIPs as well as to include “safe” variables and constraints (i.e., where uncertainties do not affect certain variables or constraints). This allows to apply the RRF to many applications since safe variables and constraints exist in most applications. We also provide first methods for computing the RRF of LPs as well as of MIPs with safe variables and constraints. Finally, we show that the new methodologies can be successfully applied to the instances in the MIPLIB 2017 for computing the RRF.

The first part of this paper studies a specific class of uncertain quadratic and linear programs, where the uncertainty enters the constraints in an affine manner and the uncertainty set is a poly-tope. It is shown that one can convert the resulting semi-infinite optimization problem into a standard QP or LP with a finite number of decision variables and a finite number of constraints. This transformation is achieved in a computationally tractable way by solving as many LPs as there are constraints in the optimization problem without uncertainty. It is also shown that if the uncertainty set is given by upper and lower bounds only, then one need not solve any LPs in order to do this transformation; computing the 1-norms of the rows of the matrix by which the uncertainty enters the constraints is sufficient. The second part of the paper reviews and extends some definitions and results on input-to-state stability for nonlinear discrete-time systems. The third part of the paper shows how one can translate a class of robust finite-horizon optimal control problems (RFHOCPs) into the class of robust convex optimization problems that was studied in the first part of the paper. It is assumed that the system under consideration is linear, that there is a persistent, but bounded state disturbance that assumes values in a polytope and that there are mixed affine constraints on the input and state. By using the results from the previous sections, it is shown that one can set up a receding horizon controller (RHC) that is input-tostate stable (ISS) and guarantees robust constraint satisfaction for all time. The number of decision variables and constraints in the RFHOCP that define the RHC law increases linearly with the horizon length.

Purpose: To improve IMRT treatments for lung cancer patients who exhibit breathing motion uncertainty; to demonstrate that combining adaptive radiation therapy with robust optimization can lead to simultaneous improvements in tumor coverage and healthy tissue sparing over non‐adaptive robust optimization methods. Methods: A model of breathing motion uncertainty was derived from previous robust optimization research. We then developed two algorithms ‐ exponential smoothing and running average ‐ to up‐date this model from fraction to fraction, using a sequence of retrospective motion probability mass functions (PMFs) from real patients. A robust optimization problem was solved for each fraction with an updated uncertainty model to generate a sequence of treatments. Delivery of the entire treatment was simulated and dose was accumulated according to the sequence of PMFs. The breathing motion model was created and updated in MATLAB, while CPLEX was used to solve the corresponding robust optimization problem as a linear program. The adaptive robust treatment was then compared with a robust treatment in which the uncertainty set was not updated over the treatment course. Results: Our adaptive robust optimization method exhibited a simultaneous improvement over non‐adaptive robust methods in both tumor coverage and healthy tissue sparing. The adaptive robust treatments were insensitive to the choice of the initial uncertainty model, and also closely comparable to idealized treatments that would be obtained with perfect foresight — i.e., treatments created with a priori knowledge of the entire sequence of future patient PMFs. Conclusions: By combining adaptive radiation therapy and robust optimization, our method combats both the instantaneous breathing motion uncertainty realized in each fraction and changes in the uncertainty that occur from fraction to fraction. Our method demonstrates that it is possible escalate dose to the tumor from the level possible with current robust methods without sacrificing healthy tissue sparing.

This paper is concerned with an uncertain multiobjective optimization problem, where the objective and constraint functions are sums-of-squares--convex (SOS-convex) polynomials, and the uncertainty sets are spectrahedra. Using a robust optimization approach, we first establish necessary and sufficient optimality conditions for weakly efficiencies of the corresponding robust multiobjective optimization problem. These optimality criteria are expressed in terms of sums-of-squares conditions and linear matrix inequalities, which provide a numerically checkable certificate of the solvability of the given optimality conditions. We then propose a dual multiobjective problem by means of sums-of-squares and linear matrix inequalities to the robust multiobjective SOS-convex polynomial optimization problem and examine weak, strong, and converse duality relations between them. In addition, we consider a semidefinite linear programming (SDP) weighted-sum relaxation problem for verifying weighted-sum efficient values of the primal problem.

In this paper, we establish strong duality between affinely adjustable two-stage robust linear programs and their dual semidefinite programs under a general uncertainty set, that covers most of the commonly used uncertainty sets of robust optimization. This is achieved by first deriving a new version of Farkas’ lemma for a parametric linear inequality system with affinely adjustable variables. Our strong duality theorem not only shows that the primal and dual program values are equal, but also allows one to find the value of a two-stage robust linear program by solving a semidefinite linear program. In the case of an ellipsoidal uncertainty set, it yields a corresponding strong duality result with a second-order cone program as its dual. To illustrate the efficacy of our results, we show how optimal storage cost of an adjustable two-stage lot-sizing problem under a ball uncertainty set can be found by solving its dual semidefinite program, using a commonly available software.

We show that adjustable robust linear programs with affinely adjustable box data uncertainties under separable polynomial decision rules admit exact sums of squares (SOS) polynomial reformulations. These problems share the same optimal values and admit a one-to-one correspondence between the optimal solutions. A sum of squares representation of non-negativity of a separable non-convex polynomial over a box plays a key role in the reformulation. This reformulation allows us to find adjustable robust solutions of uncertain linear programs under box data uncertainty by numerically solving their associated equivalent SOS polynomial optimization problem using semi-definite linear programming. We illustrate how the quality of the adjustable robust solution of a robust optimization problem with polynomial decision rules improves as the degree of the polynomial increases. Our results demonstrate that the adjustable robust solutions approach the actual optimal solution as the degree of the polynomial increases from one to fifteen.