Convex Semi-infinite Programming Research Articles

Characterization of solution sets of geodesic convex semi-infinite programming on Riemannian manifolds

Characterization of solution sets of geodesic convex semi-infinite programming on Riemannian manifolds

Persistently Feasible Robust Safe Control by Safety Index Synthesis and Convex Semi-Infinite Programming

Model mismatches prevail in real-world applications. Ensuring safety for systems with uncertain dynamic models is critical. However, existing robust safe controllers may not be realizable when control limits exist. And existing methods use loose over-approximation of uncertainties, leading to conservative safe controls. To address these challenges, we propose a control-limits aware robust safe control framework for bounded state-dependent uncertainties. We propose safety index synthesis to find a robust safe controller guaranteed to be realizable under control limits. And we solve for robust safe control via Convex Semi-Infinite Programming, which is the tightest formulation for convex bounded uncertainties and leads to the least conservative control. In addition, we analyze when and how safety can be preserved under unmodeled uncertainties. Experiment results show that our robust safe controller is always realizable under control limits and is much less conservative than strong baselines.

Control-Theoretic Splines under Uncountably Many Constraints: Fast and Exact Solutions

A constrained control-theoretic spline is a smoothing spline, which can be modeled as the output of a linear and time-invariant dynamical system with constraints on the output curve. The problem addressed in this article is that of synthesizing control-theoretic splines under uncountably many such constraints. While the standard formulation of control-theoretic splines involves finitely many constraints and has been studied from various perspectives, here we address the problem under uncountably many constraints by translating the problem into a convex semi-infinite program (SIP) and employing recently developed results to arrive at constructive and algorithmically tractable optimal solutions. We provide several examples to illustrate our results.

E-optimum sensor selection for estimation of subsets of parameters

The design of a network of observation locations is addressed in the setting of estimating unknown parameters of a spatiotemporal system modelled by a partial differential equation. Interest is in estimating only a subset of these parameters as accurately as possible. The other parameters, called nuisance parameters, must also be estimated although we are interested in neither their values, nor accuracies. The maximal eigenvalue of the covariance matrix of the maximum-likelihood estimator of the parameters of interest is used as the measure of the identification accuracy. In order to make selection of a best subset of gauged sites from a possibly very large set of candidate sites computationally tractable, its convex relaxation is introduced. Two major problems to be tackled are the potential singularity of the optimal information matrix associated with all unknown parameters and the nondifferentiability of the optimality criterion. The former is settled by imposing a constraint on the minimal allowable value of the determinant of the information matrix. The latter is resolved by reformulating the problem as a convex semi-infinite programming problem whose solution is sought by solving a sequence of finite low-dimensional min–max problems using extremely efficient generalized simplicial decomposition. The excellent performance of the proposed technique is illustrated by an example involving optimal sensor node activation in a large sensor network collecting measurements to identify a moving contaminating source.

On solving a class of fractional semi-infinite polynomial programming problems

In this paper, we study a class of fractional semi-infinite polynomial programming (FSIPP) problems, in which the objective is a fraction of a convex polynomial and a concave polynomial, and the constraints consist of infinitely many convex polynomial inequalities. To solve such a problem, we first reformulate it to a pair of primal and dual conic optimization problems, which reduce to semidefinite programming (SDP) problems if we can bring sum-of-squares structures into the conic constraints. To this end, we provide a characteristic cone constraint qualification for convex semi-infinite programming problems to guarantee strong duality and also the attainment of the solution in the dual problem, which is of its own interest. In this framework, we first present a hierarchy of SDP relaxations with asymptotic convergence for the FSIPP problem whose index set is defined by finitely many polynomial inequalities. Next, we study four cases of the FSIPP problems which can be reduced to either a single SDP problem or a finite sequence of SDP problems, where at least one minimizer can be extracted. Then, we apply this approach to the four corresponding multi-objective cases to find efficient solutions.

On Solving the Convex Semi-Infinite Minimax Problems via Superlinear 𝒱𝒰 Incremental Bundle Technique with Partial Inexact Oracle

In this paper, we study convex semi-infinite programming involving minimax problems. One of the difficulties in solving these problems is that the maximum type functions are not differentiable. Due to the nonsmooth nature of the problem, we apply the special proximal bundle scheme on the basis of [Formula: see text]-decomposition theory to solve the nonsmooth convex semi-infinite minimax problems. The proposed scheme requires an evaluation within some accuracy for all the components of the objective function. Regarding the incremental method, we only need one component function value and one subgradient which are estimated to update the bundle information and produce the search direction. Under some mild assumptions, we present global convergence and local superlinear convergence of the proposed bundle method. Numerical results of several example problems are reported to show the effectiveness of the new scheme.

Subdifferential of the supremum function: moving back and forth between continuous and non-continuous settings

In this paper we establish general formulas for the subdifferential of the pointwise supremum of convex functions, which cover and unify both the compact continuous and the non-compact non-continuous settings. From the non-continuous to the continuous setting, we proceed by a compactification-based approach which leads us to problems having compact index sets and upper semi-continuously indexed mappings, giving rise to new characterizations of the subdifferential of the supremum by means of upper semicontinuous regularized functions and an enlarged compact index set. In the opposite sense, we rewrite the subdifferential of these new regularized functions by using the original data, also leading us to new results on the subdifferential of the supremum. We give two applications in the last section, the first one concerning the nonconvex Fenchel duality, and the second one establishing Fritz-John and KKT conditions in convex semi-infinite programming.

Correction to: The CoMirror algorithm with random constraint sampling for convex semi-infinite programming

This erratum is published as due to some technical error author names appeared reversed in the final publication.

The CoMirror algorithm with random constraint sampling for convex semi-infinite programming

The CoMirror algorithm, by Beck et al. (Oper Res Lett 38(6):493–498, 2010), is designed to solve convex optimization problems with one functional constraint. At each iteration, it performs a mirror-descent update using either the subgradient of the objective function or the subgradient of the constraint function, depending on whether or not the constraint violation is below some tolerance. In this paper, we combine the CoMirror algorithm with inexact cut generation to create the SIP-CoM algorithm for solving semi-infinite programming (SIP) problems. First, we provide general error bounds for SIP-CoM. Then, we propose two specific random constraint sampling schemes to approximately solve the cut generation problem for generic SIP. When the objective and constraint functions are generally convex, randomized SIP-CoM achieves an $${\mathcal {O}}(1/\sqrt{N})$$ convergence rate in expectation (in terms of the optimality gap and SIP constraint violation). When the objective and constraint functions are all strongly convex, this rate can be improved to $${\mathcal {O}}(1/N)$$ .

Subdifferential of the Supremum via Compactification of the Index Set

We give new characterizations for the subdifferential of the supremum of an arbitrary family of convex functions, dropping out the standard assumptions of compactness of the index set and upper semi-continuity of the functions with respect to the index (J. Convex Anal. 26, 299–324, 2019). We develop an approach based on the compactification of the index set, giving rise to an appropriate enlargement of the original family. Moreover, in contrast to the previous results in the literature, our characterizations are formulated exclusively in terms of exact subdifferentials at the nominal point. Fritz–John and KKT conditions are derived for convex semi-infinite programming.

A discretization algorithm for nonsmooth convex semi-infinite programming problems based on bundle methods

We propose a discretization algorithm for solving a class of nonsmooth convex semi-infinite programming problems that is based on a bundle method. Instead of employing the inexact calculation to evaluate the lower level problem, we shall carry out a discretization scheme. The discretization method is used to get a number of discretized problems which are solved by the bundle method. In particular, the subproblem used to generate a new point is independent of the number of constraints of the discretized problem. We apply a refinement-step which can be used to guarantee the convergence of the bundle method for the discretized problems as well as reduce the cost of the evaluations for the constraint functions during iteration. In addition we adopt an aggregation technique to manage the bundle information coming from previous steps. Both theoretical convergence analysis and preliminary computational results are reported. The results obtained have shown the good performance of the new algorithm.

Convex Optimization for Group Feature Selection in Networked Data

Feature selection is at the heart of machine learning, and it is effective at facilitating data interpretability and improving prediction performance by defying the curse of dimensionality. Group feature selection is often used to reveal relationships in structured data and provide better predictive power compared with the standard feature selection methods without consideration of the grouped structure. We study a group feature selection problem in networked data in which edge weights are considered as features, while each node in the network is regarded as a group feature. This problem is particularly useful in feature selection for neuroimaging data, where the data are high dimensional and the intrinsic networked structure among the features (i.e., connectivities between regions) in brain data has to be captured properly. We propose a mathematical model based on the support vector machines (SVM), which entails the ℓ0norm regularization to restrict the number of nodes (i.e., groups). To cope with the computational challenge of the ℓ0norm regularization, we develop a convex relaxation reformulation of the proposed model as a convex semiinfinite programming (SIP). We then introduce a new iterative algorithm that achieves an optimal solution for this convex SIP. Experimental results for synthetic and real brain network data sets show that our approach gives better predictive performance compared with the state-of-the-art group feature selection and the standard feature selection methods. Our technique additionally yields a sparse subnetwork solution that is easier to interpret than those obtained by other methods.

Karush–Kuhn–Tucker optimality conditions and duality for convex semi-infinite programming with multiple interval-valued objective functions

This paper deals with convex semi-infinite programming with multiple interval-valued objective functions. We first investigate necessary and sufficient Karush–Kuhn–Tucker optimality conditions for some types of optimal solutions. Then, we formulate types of Mond–Weir and Wolfe dual problems and explore duality relations under convexity assumptions. Some examples are provided to illustrate the advantages of our results in some cases.

Algorithmic determination of immobile indices in convex SIP problems with polyhedral index sets

The concepts of immobile indices and their immobility orders are objective and important characteristics of feasible sets of semi-infinite programming (SIP) problems. They can be used for the formulation of new efficient optimality conditions without constraint qualifications. Given a class of convex SIP problems with polyhedral index sets, we describe and justify a finite constructive algorithm (algorithm DIIPS) that allows to find in a finite number of steps all immobile indices and the corresponding immobility orders along the feasible directions. This algorithm is based on a representation of the cones of feasible directions in the polyhedral index sets in the form of linear combinations of extremal rays and on the approach proposed in our previous papers for the cases of immobile indices’ sets of simpler structures. A constructive procedure of determination of the extremal rays is described, and an example illustrating the application of the DIIPS algorithm is provided.

Optimality conditions for linear copositive programming problems with isolated immobile indices

ABSTRACTIn the present paper, we apply our recent results on optimality for convex semi-infinite programming to a problem of linear copositive programming (LCP). We prove explicit optimality conditions that use concepts of immobile indices and their immobility orders and do not require the Slater constraint qualification to be satisfied. The only assumption that we impose here is that the set of immobile indices consists of isolated points and hence is finite. This assumption is weaker than the Slater condition; therefore, the optimality conditions obtained in the paper are more general when compared with those usually used in LCP. We present an example of a problem in which the new optimality conditions allow one to test the optimality of a given feasible solution while the known optimality conditions fail to do this. Further, we use the immobile indices to construct a pair of regularized dual copositive problems and show that regardless of whether the Slater condition is satisfied or not, the duality gap between the optimal values of these problems is zero. An example of a problem is presented for which the standard strict duality fails, but the duality gap obtained by using the regularized dual problem vanishes.

Semi-infinite minimax fractional programming under (Φ, <i>ρ</i>)-<i>V</i>-invexity and generalised (Φ, <i>ρ</i>)-<i>V</i>-invexity. Optimality

Semi–infinite minimax fractional programming problems with both inequality and equality constraints are considered. The sets of parametric sufficient optimality conditions for a new class of non–convex differentiable semi–infinite minimax fractional programming problems are obtained under (generalised) (Φ, ρ)–V–invexity assumptions imposed on the objective and constraint functions. With the reference to the said functions, we extend some results of optimality for a larger class of non–convex semi–infinite minimax fractional programming problems in comparison to those ones, previously established in the literature under other generalised convex notions.

Estimación de Sincrofasores en Redes Eléctricas Inteligentes: de Modelos a Restricciones de Diseño

Presentamos en primer lugar una reseña de dos populares clases de métodos de estimación de sincrofasores. Ellas son la clase de métodos basados en la transformada de Fourier discreta (DFT) y la transformada de Taylor-Fourier (TFT), que surgen de un modelo de fasor constante y polinómico, respectivamente. Se exponen las limitaciones de estos métodos, que surgen por los propios modelos asumidos, lo cual los hace poco apropiados en situaciones donde el fasor no puede representarse en términos tan simples. Este problema puede resolverse con los estimadores basados en optimización convexa semi-infinita (CSIP), un enfoque novedoso que también es descripto. En particular, enfatizamos las restricciones asociadas con las situaciones más críticas para los métodos basados en modelos, mostrando cómo controlar precisamente el desempeño de los estimadores en dichos casos. Finalmente, mostramos que los estimadores DFT y TFT son instancias particulares del estimador CSIP, de modo existe una conexión entre estos dos enfoques aparentemente diferentes. Esto abre la puerta para futuros análisis y desarrollos.

Optimality Conditions for Convex Semi-infinite Programming Problems with Finitely Representable Compact Index Sets

In the present paper, we analyze a class of convex semi-infinite programming problems with arbitrary index sets defined by a finite number of nonlinear inequalities. The analysis is carried out by employing the constructive approach, which, in turn, relies on the notions of immobile indices and their immobility orders. Our previous work showcasing this approach includes a number of papers dealing with simpler cases of semi-infinite problems than the ones under consideration here. Key findings of the paper include the formulation and the proof of implicit and explicit optimality conditions under assumptions, which are less restrictive than the constraint qualifications traditionally used. In this perspective, the optimality conditions in question are also compared to those provided in the relevant literature. Finally, the way to formulate the obtained optimality conditions is demonstrated by applying the results of the paper to some special cases of the convex semi-infinite problems.

An exchange method with refined subproblems for convex semi-infinite programming problems

In this paper, we propose a new exchange method for solving convex semi-infinite programming problems (SIPs). The traditional exchange method solves a sequence of finitely relaxed subproblems, that is, subproblems with finitely many constraints chosen from the original constraints. On the other hand, our exchange method solves a sequence of new subproblems, in which the traditional finite subproblems arerefined by the quadratic approximation. Under mild assumptions, the refined subproblems approximate the original SIP more precisely than the traditional subproblems. Moreover, although those subproblems are still SIPs, they can be solved efficiently by reformulating them as certain optimization problems with finitely many constraints. We establish the global convergence property of the proposed algorithm. Finally, we examine the efficiency of the algorithm by some numerical experiments.

Computation Algorithm for Convex Semi-infinite Program with Second-Order Cones: Special Analyses for Affine and Quadratic Case

We focus on the convex semi-infinite program with second-order cone constraints (for short, SOCCSIP), which has wide applications such as filter design, robust optimization, and so on. For solving the SOCCSIP, we propose an explicit exchange method, and prove that the algorithm terminates in a finite number of iterations. In the convergence analysis, we do not need to use the special structure of second-order cone (SOC) when the objective or constraint function is strictly convex. However, if both of them are non-strictly convex and constraint function is affine or quadratic, then we have to utilize the SOC complementarity conditions and the spectral factorization techniques associated with Euclidean Jordan algebra. We also show that the obtained output is an approximate optimum of SOCCSIP. We report some numerical results involving the application to the robust optimization in the classical convex semi-infinite program.