Linear Semi-infinite Optimization Research Articles

New applications of linear semi-infinite optimization theory in copositive optimization

This paper provides a review and update of the theory of linear semi-infinite optimization, followed by its direct application to derive existence theorems, results on the geometry of the feasible set, duality theorems, and optimality conditions for copositive optimization problems.

Computing T-optimal designs via nested semi-infinite programming and twofold adaptive discretization

Modelling real processes often results in several suitable models. In order to be able to distinguish, or discriminate, which model best represents a phenomenon, one is interested, e.g., in so-called T-optimal designs. These consist of the (design) points from a generally continuous design space at which the models deviate most from each other under the condition that they are best fitted to those points. Thus, the T-criterion represents a bi-level optimization problem, which can be transferred into a semi-infinite one but whose solution is very unstable or time consuming for non-linear models and non-convex lower- and upper-level problems. If one considers only a finite number of possible design points, a numerically well tractable linear semi-infinite optimization problem arises. Since this is only an approximation of the original model discrimination problem, we propose an algorithm which alternately and adaptively refines discretizations of the parameter as well as of the design space and, thus, solves a sequence of linear semi-infinite programs. We prove convergence of our method and its subroutine and show on the basis of discrimination tasks from process engineering that our approach is stable and can outperform the known methods.

Model-Free Bounds for Multi-Asset Options Using Option-Implied Information and Their Exact Computation

We consider derivatives written on multiple underlyings in a one-period financial market, and we are interested in the computation of model-free upper and lower bounds for their arbitrage-free prices. We work in a completely realistic setting, in that we only assume the knowledge of traded prices for other single- and multi-asset derivatives and even allow for the presence of bid–ask spread in these prices. We provide a fundamental theorem of asset pricing for this market model, as well as a superhedging duality result, that allows to transform the abstract maximization problem over probability measures into a more tractable minimization problem over vectors, subject to certain constraints. Then, we recast this problem into a linear semi-infinite optimization problem and provide two algorithms for its solution. These algorithms provide upper and lower bounds for the prices that are ε-optimal, as well as a characterization of the optimal pricing measures. These algorithms are efficient and allow the computation of bounds in high-dimensional scenarios (e.g., when d = 60). Moreover, these algorithms can be used to detect arbitrage opportunities and identify the corresponding arbitrage strategies. Numerical experiments using both synthetic and real market data showcase the efficiency of these algorithms, and they also allow understanding of the reduction of model risk by including additional information in the form of known derivative prices. This paper was accepted by Chung Piaw Teo, optimization. Funding: This work was supported by the Nanyang Technological University [NAP Grant] and the Hellenic Foundation for Research and Innovation [Grant HFRI-FM17-2152]. Supplemental Material: The data files and online appendices are available at https://doi.org/10.1287/mnsc.2022.4456 .

Isolated Calmness and Sharp Minima via Hölder Graphical Derivatives

The paper utilizes Hölder graphical derivatives for characterizing Hölder strong subregularity, isolated calmness and sharp minimum. As applications, we characterize Hölder isolated calmness in linear semi-infinite optimization and Hölder sharp minimizers of some penalty functions for constrained optimization.

Relaxed Lagrangian duality in convex infinite optimization: Reverse strong duality and optimality

We associate with each convex optimization problem posed on some locally convex space with an infinite index set T, and a given non-empty family H formed by finite subsets of T, a suitable Lagrangian-Haar dual problem. We provide reverse H-strong duality theorems, H-Farkas type lemmas and optimality theorems. Special attention is addressed to infinite and semi-infinite linear optimization problems.

Volumetric Barrier Cutting Plane Algorithms for Stochastic Linear Semi-Infinite Optimization

In this paper, we study the two-stage stochastic linear semi-infinite programming with recourse to handle uncertainty in data defining (deterministic) linear semi-infinite programming. We develop and analyze volumetric barrier cutting plane interior point methods for solving this class of optimization problems, and present a complexity analysis of the proposed algorithms. We establish our convergence analysis by showing that the volumetric barrier associated with the recourse function of stochastic linear semi-infinite programs is a strongly self-concordant barrier and forms a self-concordant family on the first-stage solutions. The dominant terms in the complexity expressions obtained in this paper are given in terms of the problem dimension and the number of realizations. The novelty of our algorithms lies in their ability to kill the effect of the radii of the largest Euclidean balls contained in the feasibility sets on the dominant complexity terms.

Optimal Robust Ordering Quantity for a New Product Under Environmental Constraints

Rapid new product introduction is an effective method for firms to capture market share. Considering the growing concerns of environmental threats, firms are under intense pressure to measure their impacts on the environment. Before launching a new product, a firm should make the appropriate decision regarding the order quantity with incomplete information to achieve the goals of both profit maximization and environmental threat minimization simultaneously. In this paper, distribution-free news vendor models with the environmental constraints are proposed to obtain efficient order decisions. The models are formulated based on the absolute minimax regret criterion. Based on the moment bound problem framework, the models are translated into a semi-infinite linear optimization problem. In addition, the closed-form solutions of the robust order quantities and carbon emission volumes are derived. The characteristics of effective order decisions are discussed under three types of carbon emissions regulations (i.e., the carbon capacity policy, carbon tax scheme, and carbon trading mechanism). By analyzing the product characteristics from both economic and environmental aspects, four possible new product introduction strategies are proposed that the firm might employ under different environmental constraints. The characteristics of the optimal order quantity are also investigated under three carbon emissions regulations across four new product introduction strategies.

Recent contributions to linear semi-infinite optimization: an update

This paper reviews the state-of-the-art in the theory of deterministic and uncertain linear semi-infinite optimization, presents some numerical approaches to this type of problems, and describes a selection of recent applications in a variety of fields. Extensions to related optimization areas, as convex semi-infinite optimization, linear infinite optimization, and multi-objective linear semi-infinite optimization, are also commented.

Recent contributions to linear semi-infinite optimization

This paper reviews the state-of-the-art in the theory of deterministic and uncertain linear semi-infinite optimization, presents some numerical approaches to this type of problems, and describes a selection of recent applications in a variety of fields. Extensions to related optimization areas, as convex semi-infinite optimization, linear infinite optimization, and multi-objective linear semi-infinite optimization, are also commented.

A generalized Farkas lemma with a numerical certificate and linear semi-infinite programs with SDP duals

In this paper, we establish a new non-homogeneous Farkas lemma for a linear semi-infinite inequality system, where the dual statement is given in terms of linear matrix inequalities and thus, it can be numerically verified by solving a semidefinite linear program. It is achieved by way of employing a special variable transformation together with a strong separation theorem. Consequently, we establish a duality theorem for a class of parametric linear semi-infinite programs which admit semi-definite linear programming (SDP) dual problems. A numerical example is given to show how a parametric linear semi-infinite optimization problem can be solved by way of solving its SDP dual problem using the Matlab toolbox CVX.

Boundary of subdifferentials and calmness moduli in linear semi-infinite optimization

This paper was originally motivated by the problem of providing a point-based formula (only involving the nominal data, and not data in a neighborhood) for estimating the calmness modulus of the optimal set mapping in linear semi-infinite optimization under perturbations of all coefficients. With this aim in mind, the paper establishes as a key tool a basic result on finite-valued convex functions in the $$n$$ -dimensional Euclidean space. Specifically, this result provides an upper limit characterization of the boundary of the subdifferential of such a convex function. When applied to the supremum function associated with our constraint system, this characterization allows us to derive an upper estimate for the aimed calmness modulus in linear semi-infinite optimization under the uniqueness of nominal optimal solution.

Calmness Modulus of Linear Semi-infinite Programs

Our main goal is to compute or estimate the calmness modulus of the argmin mapping of linear semi-infinite optimization problems under canonical perturbations, i.e., perturbations of the objective function together with continuous perturbations of the right-hand side of the constraint system (with respect to an index ranging in a compact Hausdorff space). Specifically, we provide a lower bound on the calmness modulus for semi-infinite programs with unique optimal solution which turns out to be the exact modulus when the problem is finitely constrained. The relationship between the calmness of the argmin mapping and the same property for the (sub)level set mapping (with respect to the objective function), for semi-infinite programs and without requiring the uniqueness of the nominal solution, is explored, too, providing an upper bound on the calmness modulus of the argmin mapping. When confined to finitely constrained problems, we also provide a computable upper bound as it only relies on the nominal data and parameters, not involving elements in a neighborhood. Illustrative examples are provided.

Calmness of the Argmin Mapping in Linear Semi-Infinite Optimization

This paper characterizes the calmness property of the argmin mapping in the framework of linear semi-infinite optimization problems under canonical perturbations; i.e., continuous perturbations of the right-hand side of the constraints (inequalities) together with perturbations of the objective function coefficient vector. This characterization is new for semi-infinite problems without requiring uniqueness of minimizers. For ordinary (finitely constrained) linear programs, the calmness of the argmin mapping always holds, since its graph is piecewise polyhedral (as a consequence of a classical result by Robinson). Moreover, the so-called isolated calmness (corresponding to the case of unique optimal solution for the nominal problem) has been previously characterized. As a key tool in this paper, we appeal to a certain supremum function associated with our nominal problem, not involving problems in a neighborhood, which is related to (sub)level sets. The main result establishes that, under Slater constraint qualification, perturbations of the objective function are negligible when characterizing the calmness of the argmin mapping. This result also states that the calmness of the argmin mapping is equivalent to the calmness of the level set mapping.

An Interior Point Constraint Generation Algorithm for Semi-Infinite Optimization with Health-Care Application

We propose an interior point constraint generation (IPCG) algorithm for semi-infinite linear optimization (SILO) and prove that the algorithm converges to an ε-solution of SILO after a finite number of constraints is generated. We derive a complexity bound on the number of Newton steps needed to approach the updated μ-center after adding multiple violated constraints and a complexity bound on the total number of constraints that is required for the overall algorithm to converge. We implement our algorithm to solve the sector duration optimization problem arising in Leksell Gamma Knife® Perfexion™ (Elekta, Stockholm Sweden) treatment planning, a highly specialized treatment for brain tumors. Using real patient data provided by the Department of Radiation Oncology at Princess Margaret Hospital in Toronto, Ontario, Canada, we show that our algorithm can efficiently handle problems in real-life health-care applications by providing a quality treatment plan in a timely manner. Comparing our computational results with MOSEK, a commercial software package, we show that the IPCG algorithm outperforms the classical primal-dual interior point methods on sector duration optimization problem arising in Perfexion™ treatment planning. We also compare our results with that of a projected gradient method. In both cases we show that IPCG algorithm obtains a more accurate solution substantially faster.

Comments on: Stability in linear optimization and related topics. A personal tour

Linear semi-infinite optimization problems (LSIPs) arise in a large number of applications, including functional approximation, robust optimization, optimal control, and duality theory. Whereas in many applications the index set of these inequality constraints is a compact subset of some Euclidean space, the presented survey takes an even more general point of view, where the index set is not equipped with any kind of structure. For the numerical treatment of an optimization problem it is crucial that the problem is stable under perturbations of its defining data. Different stability properties lead to a list of notions for ‘good behavior’ of an LSIP, including semicontinuity properties of feasible and optimal point mappings, different types of ill-posedness, distance to ill-posedness, as well as Lipschitz and metric regularity properties. Marco Lopez and his coauthors have studied these qualitative and quantitative notions extensively during the last fifteen years. Their remarkable results are compiled in the present, comprehensive survey about stability in linear semi-infinite optimization. Since the main questions in the framework of this survey appear to be answered, I would like to take this opportunity and suggest to slightly extend this setting. In fact, while the index set T itself is assumed to be fixed in the present approach, there are at least three possible ways to study LSIPs with parametric index sets.

On stable uniqueness in linear semi-infinite optimization

This paper is intended to provide conditions for the stability of the strong uniqueness of the optimal solution of a given linear semi-infinite optimization (LSIO) problem, in the sense of maintaining the strong uniqueness property under sufficiently small perturbations of all the data. We consider LSIO problems such that the family of gradients of all the constraints is unbounded, extending earlier results of Nürnberger for continuous LSIO problems, and of Helbig and Todorov for LSIO problems with bounded set of gradients. To do this we characterize the absolutely (affinely) stable problems, i.e., those LSIO problems whose feasible set (its affine hull, respectively) remains constant under sufficiently small perturbations.

Bounding Contingent Claim Prices via Hedging Strategy with Coherent Risk Measures

We generalize the notion of arbitrage based on the coherent risk measure, and investigate a mathematical optimization approach for tightening the lower and upper bounds of the price of contingent claims in incomplete markets. Due to the dual representation of coherent risk measures, the lower and upper bounds of price are located by solving a pair of semi-infinite linear optimization problems, which further reduce to linear optimization when conditional value-at-risk (CVaR) is used as risk measure. We also show that the hedging portfolio problem is viewed as a robust optimization problem. Tuning the parameter of the risk measure, we demonstrate by numerical examples that the two bounds approach to each other and converge to a price that is fair in the sense that seller and buyer face the same amount of risk.

Semi-Infinite Optimization under Convex Function Perturbations: Lipschitz Stability

This paper is devoted to the study of the stability of the solution map for the parametric convex semi-infinite optimization problem under convex function perturbations in short, PCSI. We establish sufficient conditions for the pseudo-Lipschitz property of the solution map of PCSI under perturbations of both objective function and constraint set. The main result obtained is new even when the problems under consideration reduce to linear semi-infinite optimization. Examples are given to illustrate the obtained results.

Error bounds for the inverse feasible set mapping in linear semi-infinite optimization via a sensitivity dual approach

In this article, some sensitivity analysis of the dual optimal value in linear semi-infinite optimization is carried out via the notion of primal/dual asymptotic solution. The sensitivity results are then applied to derive some Hoffman-type inequalities (error bounds). Like in [Renegar, J., 1994, Some perturbation theory for linear programming. Mathematical Programming, 65A, 73–91], asymptotic solutions also turn out to be a key tool for any sensitivity analysis in the setting of semi-infinite linear duality.

On the Lipschitz Modulus of the Argmin Mapping in Linear Semi-Infinite Optimization

This paper is devoted to quantify the Lipschitzian behavior of the optimal solutions set in linear optimization under perturbations of the objective function and the right hand side of the constraints (inequalities). In our model, the set indexing the constraints is assumed to be a compact metric space and all coefficients depend continuously on the index. The paper provides a lower bound on the Lipschitz modulus of the optimal set mapping (also called argmin mapping), which, under our assumptions, is single-valued and Lipschitz continuous near the nominal parameter. This lower bound turns out to be the exact modulus in ordinary linear programming, as well as in the semi-infinite case under some additional hypothesis which always holds for dimensions n ⩽ 3. The expression for the lower bound (or exact modulus) only depends on the nominal problem’s coefficients, providing an operative formula from the practical side, specially in the particular framework of ordinary linear programming, where it constitutes the sharp Lipschitz constant. In the semi-infinite case, the problem of whether or not the lower bound equals the exact modulus for n > 3 under weaker hypotheses (or none) remains as an open problem.