Exact dual semi-definite programs for affinely adjustable robust SOS-convex polynomial optimization problems

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.

We develop a modular and tractable framework for solving an adaptive distributionally robust linear optimization problem, where we minimize the worst-case expected cost over an ambiguity set of probability distributions. The adaptive distributionally robust optimization framework caters for dynamic decision making, where decisions adapt to the uncertain outcomes as they unfold in stages. For tractability considerations, we focus on a class of second-order conic (SOC) representable ambiguity set, though our results can easily be extended to more general conic representations. We show that the adaptive distributionally robust linear optimization problem can be formulated as a classical robust optimization problem. To obtain a tractable formulation, we approximate the adaptive distributionally robust optimization problem using linear decision rule (LDR) techniques. More interestingly, by incorporating the primary and auxiliary random variables of the lifted ambiguity set in the LDR approximation, we can significantly improve the solutions, and for a class of adaptive distributionally robust optimization problems, exact solutions can also be obtained. Using the new LDR approximation, we can transform the distributionally adaptive robust optimization problem to a classical robust optimization problem with an SOC representable uncertainty set. Finally, to demonstrate the potential for solving management decision problems, we develop an algebraic modeling package and illustrate how it can be used to facilitate modeling and obtain high-quality solutions for medical appointment scheduling and inventory management problems. The electronic companion is available at https://doi.org/10.1287/mnsc.2017.2952 . This paper was accepted by Noah Gans, optimization.

A convergent hierarchy of SDP relaxations for a class of hard robust global polynomial optimization problems

This paper studies robust solutions and semidefinite linear programming (SDP) relaxations of a class of convex polynomial programs in the face of data uncertainty. The class of convex programs, called robust SOS-convex programs, includes robust quadratically constrained convex programs and robust separable convex polynomial programs. It establishes sums of squares polynomial representations characterizing robust solutions and exact SDP-relaxations of robust SOS-convex programs under various commonly used uncertainty sets. In particular, the results show that the polytopic and ellipsoidal uncertainty sets, that allow second-order cone re-formulations of robust quadratically constrained programs, continue to permit exact SDP-relaxations for a broad class of robust SOS-convex programs. They also yield exact second-order cone relaxation for robust quadratically constrained programs.

Robust optimization is a common optimization framework under uncertainty when problem parameters are unknown, but it is known that they belong to some given uncertainty set. In the robust optimization framework, a min-max problem is solved wherein a solution is evaluated according to its performance on the worst possible realization of the parameters. In many cases, a straightforward solution to a robust optimization problem of a certain type requires solving an optimization problem of a more complicated type, which might be NP-hard in some cases. For example, solving a robust conic quadratic program, such as those arising in a robust support vector machine (SVM) with an ellipsoidal uncertainty set, leads in general to a semidefinite program. In this paper, we develop a method for approximately solving a robust optimization problem using tools from online convex optimization, where at every stage a standard (nonrobust) optimization program is solved. Our algorithms find an approximate robust solution using a number of calls to an oracle that solves the original (nonrobust) problem that is inversely proportional to the square of the target accuracy.

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.

Quadratically adjustable robust linear optimization with inexact data via generalized S-lemma: Exact second-order cone program reformulations

Optimization is a key analytical technique used for quantitative decision-making in real-world problems. In practice, many situations call for decision-making in the face of incomplete knowledge and/or dynamic environments. Making high-quality decisions in these settings requires optimization techniques that are designed to account for uncertainty. Furthermore, as new technologies are developed, more complex higher-dimensional optimization models become prevalent. This dissertation examines various models for optimization under uncertainty, as well as efficient algorithms for solving such models that are scalable as the model size grows.<br>We study three models for optimization under uncertainty: robust optimization (RO), joint estimation-optimization (JEO), and joint prediction-optimization (JPO). Robust optimization accounts for inexact information by finding solutions, which remain feasible to all perturbations of<br>inputs within a given uncertainty set. Joint estimation-optimization considers a dynamic setting where inputs are updated over time as new data is collected and converge to some ideal input that is<br>not revealed to the modeller. Joint prediction-optimization considers the use of a prediction model to obtain optimization inputs from side information, an approach that is widely used amongst practitioners.<br>The dissertation considers theoretical properties and algorithmic performance guarantees for these three models.<br>We first present a generic framework to derive primal-dual algorithms for both RO and JEO. Previously, algorithms for such models were derived in an ad-hoc manner, and analyzed on a case-by-case basis. Our framework considers both of these optimization under uncertainty models<br>through a common lens of saddle point problems. By analyzing these, we highlight three quantities which directly bound the performance guarantees for our respective models, and show how regret<br>minimization techniques from online convex optimization can be used to control these three quantities. Thus, our framework allows us to transfer regret bounds for these quantities into performance guarantees for the associated algorithms. Since regret minimization algorithms from online convex<br>optimization are key to our framework, we also examine these, and in particular derive improved regret bounds for RO and JEO in the presence of favourable structure such as strong convexity<br>and smoothness. We show that a number of previous algorithms for both robust optimization and joint estimation optimization can be derived from our uni ed framework. More importantly, our framework can be used to derive more efficient algorithms for both models in a principled manner. For robust<br>optimization, our framework is used to derive algorithms that can drastically reduce the cost of iterative methods by replacing nominal oracles with cheaper first-order updates. For joint estimation optimization, we derive algorithms for the non-smooth strongly convex setting, which has not been considered previously.<br>We demonstrate the use of our framework through two examples: robust quadratic programming with ellipsoidal uncertainty sets, and dynamic non-parametric choice model estimation. For robust quadratic programming, we analyze the trust-region subproblem (TRS). The TRS is the<br>well-studied problem of minimizing a non-convex quadratic function over the unit ball, and it arises naturally in the context of robust quadratic constraints. We give a second-order cone based convexi cation of TRS which, in contrast to previous work, is still in the space of original variables.<br>We then show how to apply this convexication to robust quadratic programming, and derive two efficient algorithms for it using our framework. We carry out a numerical study on robust portfolio optimization problems, and the numerical results show improvement of our approach over previous approaches in the high-dimensional regime. We frame dynamic non-parametric choice model estimation as an instance of JEO. A particular challenge in this setting is the high-dimensionality of the resulting primal problem. Nevertheless, our generic primal-dual framework encompassing JEO applications is quite flexible and allows us to derive algorithms that can bypass this high dimensionality challenge. We test our approach for non-parametric choice estimation computationally, and highlight interesting trade-o s between data updating and convergence rates. Finally, we give a joint analysis of prediction and optimization. A natural performance measure in this setting is the optimality gap. Unfortunately, it is difficult to directly tune prediction models using this performance measure due to its non-convexity. We thus characterize sufficient conditions under which the more common prediction performance measures arising in statistics/machine learning, such as squared error, can be related to the true optimality gap performance measure. We derive conditions on a performance measure that guarantee that the optimality gap will be minimized, and give an explicit relationship between the squared error and the optimality gap. Such conditions allow practitioners to choose prediction methods for obtaining optimization parameters in a more judicious manner.<br>

This paper focuses on the study of robust two-stage quadratic multiobjective optimization problems. We formulate new necessary and sufficient optimality conditions for a robust two-stage multiobjective optimization problem. The obtained optimality conditions are presented by means of linear matrix inequalities and thus they can be numerically validated by using a semidefinite programming problem. The proposed optimality conditions can be elaborated further as second-order conic expressions for robust two-stage quadratic multiobjective optimization problems with separable functions and ellipsoidal uncertainty sets. We also propose relaxation schemes for finding a (weak) efficient solution of the robust two-stage multiobjective problem by employing associated semidefinite programming or second-order cone programming relaxations. Moreover, numerical examples are given to demonstrate the solution variety of our flexible models and the numerical verifiability of the proposed schemes.

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.

This paper presents a new approach aims to solve robust multidisciplinary design optimization MDO problem called Improved Multi-objective Robust Collaborative Optimization . This method combines the Multi-objective Robust Collaborative Optimization method, the Worst Possible Point constraint cuts and the Genetic algorithm NSGA-II type as an optimizer to solve the robust optimization problem of complex structure named Y-stiffened panel under interval uncertainty . The proposed approach hierarchically decomposes the optimization problem into a structure level considered as an upper level in the Y-stiffened panel and a second level considered as a lower level of the studied panel. A robust multi-objective optimization problem intended to optimize the eigenfrequency, the global mass and the displacement at a fixed point of the Y-stiffened panel at the first level and each structure’s robust optimization problem allows optimizing its eigenfrequency and mass limited by their local constraint functions at the second one. Tor demonstrate our method, an engineering example of Y-stiffened panel is treated. A good performance of proposed method is proved by a comparison between obtained results and Non-Distributed Multi-objective Robust Optimization .

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 semi-definite linear programs, among them uncertain semi-definite 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 optimization (RO) is a tractable method to address uncertainty in optimization problems where uncertain parameters are modeled as belonging to uncertainty sets that are commonly polyhedral or ellipsoidal. The two most frequently described methods in the literature for solving RO problems are reformulation to a deterministic optimization problem or an iterative cutting-plane method. There has been limited comparison of the two methods in the literature, and there is no guidance for when one method should be selected over the other. In this paper we perform a comprehensive computational study on a variety of problem instances for both robust linear optimization (RLO) and robust mixed-integer optimization (RMIO) problems using both methods and both polyhedral and ellipsoidal uncertainty sets. We consider multiple variants of the methods and characterize the various implementation decisions that must be made. We measure performance with multiple metrics and use statistical techniques to quantify certainty in the results. We find for polyhedral uncertainty sets that neither method dominates the other, in contrast to previous results in the literature. For ellipsoidal uncertainty sets we find that the reformulation is better for RLO problems, but there is no dominant method for RMIO problems. Given that there is no clearly dominant method, we describe a hybrid method that solves, in parallel, an instance with both the reformulation method and the cutting-plane method. We find that this hybrid approach can reduce runtimes to 50–75 % of the runtime for any one method and suggest ways that this result can be achieved and further improved on.

Surrogate duality for robust optimization

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.