Linear Semi-infinite Optimization Problems Research Articles

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 .

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.

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.

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.

On the Continuity of the Optimal Value in Parametric Linear Optimization: Stable Discretization of the Lagrangian Dual of Nonlinear Problems

This paper is focused on the stability of the optimal value, and its immediate repercussion on the stability of the optimal set, for a general parametric family of linear optimization problems in ℝn. In our approach, the parameter ranges over an arbitrary metric space, and each parameter determines directly a set of coefficient vectors describing the linear system of constraints. Thus, systems associated with different parameters are not required to have the same number (cardinality) of inequalities. In this way, discretization techniques for solving a nominal linear semi-infinite optimization problem may be modeled in terms of suitable parametrized problems. The stability results given in the paper are applied to the stability analysis of the Lagrangian dual associated with a parametric family of nonlinear programming problems. This dual problem is translated into a linear (semi-infinite) programming problem and, then, we prove that the lower semicontinuity of the corresponding feasible set mapping, the continuity of the optimal value function, and the upper semicontinuity of the optimal set mapping are satisfied. Then, the paper shows how these stability properties for the dual problem entail a nice behavior of parametric approximation and discretization strategies (in which an ordinary linear programming problem may be considered in each step). This approximation–discretization process is formalized by means of considering a double parameter: the original one and the finite subset of indices (grid) itself. Finally, the convex case is analyzed, showing that the referred process also allows us to approach the primal problem.

Unicity Results for General Linear Semi-Infinite Optimization Problems Using a New Concept of Active Constraints

We consider parametric semi-infinite optimization problems without the usual asssumptions on the continuity of the involved mappings and on the compactness of the index set counting the inequalities. We establish a characterization of those optimization problems which have a unique or strongly unique solution and which are stable under small pertubations. This result generalizes a well-known theorem of Nurnberger. The crucial roles in our investigations are a new concept of active constraints, a generalized Slater's condition, and a Kuhn—Tucker-type theorem. Finally, we give some applications in vector optimization, for approximation problems in normed spaces, and in the stability of the minimal value.

Strong unicity and alternation for linear optimization

We consider linear semi-infinite optimization problems and prove characterizations for strong unicity. One of these is a weak alternation property. This result suggests the introduction of the property of regular strong unicity, which is equivalent to a stronger alternation property. The theorems are used in order to prove a Haar-type theorem for linear optimization problems. An application to best Chebyshev approximation is given.

Global unicity in semi-infinite optimization

A characterization of those linear semi-infinite optimization problems which have a unique resp. strongly unique solution for all restriction functions is given. This result is applied to best approximation of continuous functions. The classical theorem of Haar in Chebyshev approximation follows as a special case. More over it is shown that global unicity in one-sided L1-approximation is equivalent to the non-existence of certain quadrature formulas. Using this characterization a general non-unicity result on one-sided L1-approximation is obtained.