Systems Of Convex Inequalities Research Articles

Feasibility problems via paramonotone operators in a convex setting

This paper is focussed on some properties of paramonotone operators on Banach spaces and their application to certain feasibility problems for convex sets in a Hilbert space and convex systems in the Euclidean space. In particular, it shows that operators that are simultaneously paramonotone and bimonotone are constant on their domains, and this fact is applied to tackle two particular situations. The first one, closely related to simultaneous projections, deals with a finite amount of convex sets with an empty intersection and tackles the problem of finding the smallest perturbations (in the sense of translations) of these sets to reach a nonempty intersection. The second is focussed on the distance to feasibility; specifically, given an inconsistent convex inequality system, our goal is to compute/estimate the smallest right-hand side perturbations that reach feasibility. We advance that this work derives lower and upper estimates of such a distance, which become the exact value when confined to linear systems.

Formulas of first-ordered and second-ordered generalization differentials for convex robust systems with applications

In the paper, we start by establishing first and second-ordered analysis for a convex inequality system that contains uncertainty data, including calculating normal and tangent cones, second-ordered tangent sets for the solution set to this system, and first and second-ordered epi-subderivatives for the indicator function of its solution set. Then, we provide second-ordered necessary and sufficient optimality conditions for strict solutions of convex robust optimization problems. Moreover, an associated algorithm converging quickly to a solution for the class of quadratic robust optimization problems is proposed. The theoretical results are newly obtained under weak qualification conditions, and numerical examples show the advantage of the given method.

Robust error bounds for uncertain convex inequality systems with applications

In this paper, we devote ourselves to studying robust error bounds for a convex inequality system in the face of data uncertainty. Under the assumption that uncertain sets are convex compact, we present a sufficient condition for the existence of robust error bounds of the uncertain convex inequality system by means of the associated recession cone and recession functions. When the uncertain sets are only compact, we establish a necessary and sufficient condition for the uncertain convex inequality system to possess a robust error bound by the Ekeland variational principle. As an immediate application, we obtain robust error bounds for the uncertain convex polynomial inequality system.

Formulas and applications for the normal cone to the solution set of an infinite convex inequality system with a constraint

Formulas and applications for the normal cone to the solution set of an infinite convex inequality system with a constraint

Error bounds for inequality systems defining convex sets

The main goal in this paper is to devise an approach to explicitly calculate the constant in the Hoffman’s error bound for (not necessarily convex) inequality systems defining convex sets. We give a constructive proof of the Hoffman’s error bound and show that we can use our method to calculate the constant at least in simple cases.

Lipschitz modulus of linear and convex inequality systems with the Hausdorff metric

This paper analyzes the Lipschitz behavior of the feasible set mapping associated with linear and convex inequality systems in {mathbb {R}}^{n}. To start with, we deal with the parameter space of linear (finite/semi-infinite) systems identified with the corresponding sets of coefficient vectors, which are assumed to be closed subsets of {mathbb {R}} ^{n+1}. In this framework the size of perturbations is measured by means of the (extended) Hausdorff distance. A direct antecedent, extensively studied in the literature, comes from considering the parameter space of all linear systems with a fixed index set, T, where the Chebyshev (extended) distance is used to measure perturbations. In the present work we propose an appropriate indexation strategy which allows us to establish the equality of the Lipschitz moduli of the feasible set mappings in both parametric contexts, as well as to benefit from existing results in the Chebyshev setting for transferring them to the Hausdorff one. In a second stage, the possibility of perturbing directly the set of coefficient vectors of a linear system leads to new contributions on the Lipschitz behavior of convex systems via linearization techniques.

Robust Farkas-Minkowski Constraint Qualification for Convex Inequality System Under Data Uncertainty

The Farkas-Minkowski constraint qualification is an important concept within the theory and applications of mathematical programs with inequality constraints. In this paper, we mainly deal with the robust version of Farkas-Minkowski constraint qualification for convex inequality system under data uncertainty. We prove that the existence of robust global error bound is a sufficient condition for ensuring robust Farkas-Minkowski constraint qualification for convex inequality system in face of data uncertainty, where the uncertain data belong to a prescribed compact and convex uncertainty set. Moreover, we show that the converse is true for convex quadratic inequality system, when the uncertain data belong to a scenario uncertainty set.

Radius of Robust Feasibility of System of Convex Inequalities with Uncertain Data

In this paper, we investigate the radius of robust feasibility of system of uncertain convex inequalities by the Minkowski function. We firstly establish an upper bound and a lower bound for radius of robust feasibility of the system of uncertain convex inequalities. Exact formulas of radius of robust feasibility of the system are derived under the nonsymmetric and symmetric assumptions of the uncertain sets. We also obtain a characterization on the positiveness of radius of robust feasibility for the system. Lastly, explicit tractable formulas for computing the radius of robust feasibility of the system are presented when the uncertain sets are ellipsoids, polytopes, boxes and unit ball, respectively.

Primal–Dual Optimization Conditions for the Robust Sum of Functions with Applications

This paper associates a dual problem to the minimization of an arbitrary linear perturbation of the robust sum function introduced in DOI 10.1007/s11228-019-00515-2. It provides an existence theorem for primal optimal solutions and, under suitable duality assumptions, characterizations of the primal-dual optimal set, the primal optimal set, and the dual optimal set, as well as a formula for the subdiffential of the robust sum function. The mentioned results are applied to get simple formulas for the robust sums of subaffine functions (a class of functions which contains the affine ones) and to obtain conditions guaranteeing the existence of best approximate solutions to inconsistent convex inequality systems.

Portfolio Optimization with Higher-order Stochastic Dominance Constraints

Optimal portfolio choice is investigated for investors with various types of higher-order risk aversion. Finite systems of convex inequalities are introduced to evaluate the efficiency of a given benchmark, and for constructing enhanced portfolios which dominate the benchmark for all higher-order risk averters. An analysis of equity industry rotation strategies shows that accounting for skewness love and kurtosis aversion increases long positions in recent winner industries which feature more favorable skewness and kurtosis than the market index. An analysis of equity stock index options combinations shows that accounting for higher-order risk aversion leads to buying of underpriced option series in addition to the standard solution of writing of overpriced options. In both applications, optimization with higher-order risk constraints appears leads to superior out-of-sample performance compared with simpler strategies based on heuristic rules, single utility functions and limiting portfolio variance.

Point-Based Neighborhoods for Sharp Calmness Constants in Linear Programming

From a computational point of view, this paper provides a significant advance in the study of the calmness property of ordinary (finite) linear programs under canonical perturbations (i.e., perturbations of the objective function coefficient vector and the right-hand side of the constraint system). In the recent literature we find, for both the feasible and the optimal set (argmin) mappings, computable expressions for the corresponding calmness moduli. These expressions are called point-based as far as they only depend on the nominal data. In this paper we show that both calmness moduli are indeed (sharp) calmness constants, and provide point-based expressions for neighborhoods where such constants work. We show that our results cannot be extended to general convex inequality systems under right-hand side perturbations.

Higher-degree stochastic dominance optimality and efficiency

We characterize a range of Stochastic Dominance (SD) relations by means of finite systems of convex inequalities. For ‘SD optimality’ of degree 1 to 4 and ‘SD efficiency’ of degree 2 to 5, we obtain exact systems that can be implemented using Linear Programming or Convex Quadratic Programming. For SD optimality of degree five and higher, and SD efficiency of degree six and higher, we obtain necessary conditions. We use separate model variables for the values of the derivatives of all relevant orders at all relevant outcome levels, which allows for preference restrictions beyond the standard sign restrictions. Our systems of inequalities can be interpreted in terms of piecewise polynomial utility functions with a number of pieces that increases with the number of outcomes and the degree of SD. An empirical study analyzes the relevance of higher-order risk preferences for comparing a passive stock market index with actively managed stock portfolios in standard data sets from the empirical asset pricing literature.

Theorems of the Alternative for Systems of Convex Inequalities

Systems of convex inequalities in function spaces are considered. Solvability conditions are obtained in the form of a theorem of the alternative. We revisit some results from the literature where such theorems are incorrect. We present counterexamples concerning these results and by introducing some regularity conditions we obtain new theorems which are dedicated to solvability of systems of convex inequalities.

Higher-Degree Stochastic Dominance Optimality and Efficiency

We characterize a range of Stochastic Dominance relations by means of finite systems of convex inequalities. For 'SD optimality' of degree N = 1, 2, 3, 4 and 'SD efficiency' of degree N = 2, 3, 4, 5, we obtain exact systems that can be implemented using Linear Programming or Convex Quadratic Programming. For SD optimality of degree N>=5 and SD efficiency of degree N>=6, we obtain necessary conditions. Our analysis leads to higher accuracy than existing necessary and approximate conditions for higher-degree relations if the partition of the outcomes domain is coarse. In addition, we use separate model variables for the values of the derivatives of all relevant orders at all relevant outcome levels, which allows for preference restrictions beyond the standard sign restrictions. Our systems of inequalities can be interpreted in terms of piecewise polynomial utility functions with a number of pieces that increases with the number of outcomes (T) and the degree of SD (N).

Nonlinear error bounds for quasiconvex inequality systems

The error bound is an inequality that restricts the distance from a vector to a given set by a residual function. The error bound has so many useful applications, for example in variational analysis, in convergence analysis of algorithms, in sensitivity analysis, and so on. For convex inequality systems, Lipschitzian error bounds are studied mainly. If an inequality system is not convex, it is difficult to show the existence of a Lipschitzian global error bound in general. Hence for nonconvex inequality systems, Holderian error bounds and nonlinear error bounds have been investigated. For quasiconvex inequality systems, there are so many examples such that systems do not have Lipschitzian and Holderian error bounds. However, the research of nonlinear error bounds for quasiconvex inequality systems have not been investigated yet as far as we know. In this paper, we study nonlinear error bounds for quasiconvex inequality systems. We show the existence of a global nonlinear error bound by a generator of a quasiconvex function and a constraint qualification. We show well-posedness of a quasiconvex function by the error bound.

On estimates for solutions of systems of convex inequalities

The distance from a given point to the solution set of a system of strict and nonstrict inequalities described by convex functions is estimated. As consequences, estimates for the distance from a given point to the Lebesgue set of a convex function are obtained and sufficient conditions for convex-valued set-valued mappings to be covering are established.

γ-Active Constraints in Convex Semi-Infinite Programming

In this article, we extend the definition of γ-active constraints for linear semi-infinite programming to a definition applicable to convex semi-infinite programming, by two approaches. The first approach entails the use of the subdifferentials of the convex constraints at a point, while the second approach is based on the linearization of the convex inequality system by means of the convex conjugates of the defining functions. By both these methods, we manage to extend the results on γ-active constraints from the linear case to the convex case.

Comments on: Farkas’ lemma: three decades of generalizations for mathematical optimization

The celebrated Farkas lemma originated in Farkas (1902) provides us an attractively simple and extremely useful characterization for a linear inequality to be a consequence of a linear inequality system on the Euclidean space Rn . This lemma has been extensively studied and extended in many directions, including conic systems (linear, sublinear, convex), infinite/semi-infinite convex inequality systems and some classes of nonconvex systems such as the difference of convex (DC) systems and composite convex systems, and the rich results are spread in the literature.

Strong Abadie CQ, ACQ, calmness and linear regularity

The Abadie CQ (ACQ) for convex inequality systems is a fundamental notion in optimization and approximation theory. In terms of the contingent cone and tangent derivative, we extend the Abadie CQ to more general convex multifunction cases and introduce the strong ACQ for both multifunctions and inequality systems. Some seemly unrelated notions are unified by the new ACQ and strong ACQ. Relationships among ACQ, strong ACQ, basic constraint qualification (BCQ) and strong BCQ are discussed. Using the strong ACQ, we study calmness of a closed and convex multifunction between two Banach spaces and, different from many existing dual conditions for calmness, establish several primal characterizations of calmness. As applications, some primal characterizations for error bounds and linear regularity are established; in particular, some existing results are improved.

A Subgradient-Like Algorithm for Solving Vector Convex Inequalities

In this paper, we propose a strongly convergent variant of Robinson's subgradient algorithm for solving a system of vector convex inequalities in Hilbert spaces. The advantage of the proposed method is that it converges strongly, when the problem has solutions, under mild assumptions. The proposed algorithm also has the following desirable property: the sequence converges to the solution of the problem, which lies closest to the starting point and remains entirely in the intersection of three balls with radius less than the initial distance to the solution set.