Qualification Conditions Research Articles

Revisiting some Rules of Convex Analysis

Convex analysis is devoted to the study and the use of four notions: conjugate functions, normal cones, subdifferentials, support functions under convexity assumptions. These notions are closely related and the calculus rules for one of them imply calculus rules for the other ones. But for none of these notions such rules are always valid without additional assumptions. Thus, much effort has been devoted to these “qualification conditions” (Attouch-Brezis, Boţ and his co-authors, Burachik-Jeyakumar, Rockafellar...). We introduce a new condition that requires a pointwise outer semicontinuity (or closedness) property of an appropriate multifunction. For subdifferentials (resp. normal cones), the multifunction is defined in terms of subdifferentials (resp. normal cones). We relate the new condition with previous ones from the literature.

Constraint Qualification Based Detection Method for Nodal Price Multiplicity

This paper proposes a fast, standalone method for detecting multiple dual solutions (MDS) of the market dispatch problem. Using linear independence constraint qualification conditions and matrix theory, we characterize the conditions for MDS and develop a procedure for their detection. Our procedure exploits the structure of the market dispatch problem to improve computational efficiency. We demonstrate our method on eight nodal pricing scenarios, in four test systems.

About Intrinsic Transversality of Pairs of Sets

The article continues the study of the 'regular' arrangement of a collection of sets near a point in their intersection. Such regular intersection or, in other words, transversality properties are crucial for the validity of qualification conditions in optimisation as well as subdifferential, normal cone and coderivative calculus, and convergence analysis of computational algorithms. One of the main motivations for the development of the transversality theory of collections of sets comes from the convergence analysis of alternating projections for solving feasibility problems. This article targets infinite dimensional extensions of the intrinsic transversality property introduced recently by Drusvyatskiy, Ioffe and Lewis as a sufficient condition for local linear convergence of alternating projections. Several characterisations of this property are established involving new limiting objects defined for pairs of sets. Special attention is given to the convex case.

Determinants of Intended Retirement Timing in Germany

Recent studies indicate that pension reforms in Germany might have caused social inequality in the transition from work to retirement. This paper investigates a potential emergence of social inequality in retirement transitions by focusing on future pensioners. An analysis of the data set <italic>BIBB / BAuA Employment Survey of the Working Population on Qualification and Working Conditions in Germany 2012</italic> shows that both low- and high-educated older workers plan to retire later than their medium-educated peers, whereas their reasons differ markedly. While the higher educated tend to favour a later exit from the workforce due to a strong identification with their job, the primary reason for the low-educated to postpone retirement is financial pressure. Determinanten des geplanten Renteneintritts in Deutschland Die vorliegende Studie untersucht soziale Ungleichheiten beim Rentenübergang mit einem Fokus auf das geplante Renteneintrittsalter. Basierend auf Daten der <italic>BIBB / BAuA Erwerbstätigenbefragung 2012</italic> zeigen die Analysen, dass sowohl hoch als auch niedrig gebildete ältere Arbeitnehmer und Arbeitnehmerinnen planen, später in Rente zu gehen als Vergleichsgruppen mit einem mittleren Bildungsniveau. Allerdings unterscheiden sich die Gründe für dieses Verhalten. Während die Hochgebildeten eher nicht-materiale Gründe angeben, z. B. hohe Identifikation mit ihrem Beruf, stehen für die Niedriggebildeten finanzielle Zwänge im Mittelpunkt ihrer Rentenentscheidung. JEL–Klassifizierung: H10, H55, H75

Nonlinear programming and Grossone: Quadratic Programing and the role of Constraint Qualifications

A novel and interesting approach to infinite and infinitesimal numbers was recently proposed in a series of papers and a book by Sergeyev. This novel numeral system is based on the use of a new infinite unit of measure (the number grossone, indicated by the numeral ①), the number of elements of the set, IN, of natural numbers. Based on the use of ①, De Cosmis and De Leone (2012) have then proposed a new exact differentiable penalty function for constrained optimization problems. In this paper these results are specialized to the important case of quadratic problems with linear constraints. Moreover, the crucial role of Constraint Qualification conditions (well know in constraint minimization literature) is also discussed with reference to the new proposed penalty function.

On M-stationarity conditions in MPECs and the associated qualification conditions

Depending on whether a mathematical program with equilibrium constraints (MPEC) is considered in its original or its enhanced (via KKT conditions) form, the assumed qualification conditions as well as the derived necessary optimality conditions may differ significantly. In this paper, we study this issue when imposing one of the weakest possible qualification conditions, namely the calmness of the perturbation mapping associated with the respective generalized equations in both forms of the MPEC. It is well known that the calmness property allows one to derive the so-called M-stationarity conditions. The restrictiveness of assumptions and the strength of conclusions in the two forms of the MPEC is also strongly related to the qualification conditions on the “lower level”. For instance, even under the linear independence constraint qualification (LICQ) for a lower level feasible set described by $$\mathscr {C}^1$$ functions, the calmness properties of the original and the enhanced perturbation mapping are drastically different. When passing to $$\mathscr {C}^{1,1}$$ data, this difference still remains true under the weaker Mangasarian–Fromovitz constraint qualification, whereas under LICQ both the calmness assumption and the derived optimality conditions are fully equivalent for the original and the enhanced form of the MPEC. After clarifying these relations, we provide a compilation of practically relevant consequences of our analysis in the derivation of necessary optimality conditions. The obtained results are finally applied to MPECs with structured equilibria.

Convergent conic linear programming relaxations for cone convex polynomial programs

In this paper we show that a hierarchy of conic linear programming relaxations of a cone-convex polynomial programming problem converges asymptotically under a mild well-posedness condition which can easily be checked numerically for polynomials. We also establish that an additional qualification condition guarantees finite convergence of the hierarchy. Consequently, we derive convergent semi-definite programming relaxations for convex matrix polynomial programs as well as easily tractable conic linear programming relaxations for a class of pth-order cone convex polynomial programs.

Friendly Jamming in a MIMO Wiretap Interference Network: A Nonconvex Game Approach

We consider joint optimization of artificial noise (AN) and information signals in a MIMO wiretap interference network, wherein the transmission of each link may be overheard by several MIMO-capable eavesdroppers. Each information signal is accompanied with AN, generated by the same user to confuse nearby eavesdroppers. Using a noncooperative game, a distributed optimization mechanism is proposed to maximize the secrecy rate of each link. The decision variables here are the covariance matrices for the information signals and ANs. However, the nonconvexity of each link's optimization problem (i.e., best response) makes conventional convex games inapplicable, even to find whether a Nash Equilibrium (NE) exists. To tackle this issue, we analyze the proposed game using a relaxed equilibrium concept, called quasi-Nash equilibrium (QNE). Under a constraint qualification condition for each player's problem, the set of QNEs includes the NE of the proposed game. We also derive the conditions for the existence and uniqueness of the resulting QNE. It turns out that the uniqueness conditions are too restrictive, and do not always hold in typical network scenarios. Thus, the proposed game often has multiple QNEs, and convergence to a QNE is not always guaranteed. To overcome these issues, we modify the utility functions of the players by adding several specific terms to each utility function. The modified game converges to a QNE even when multiple QNEs exist. Furthermore, players have the ability to select a desired QNE that optimizes a given social objective (e.g., sum-rate or secrecy sum-rate). Depending on the chosen objective, the amount of signaling overhead as well as the performance of resulting QNE can be controlled. Simulations show that due to the QNE selection mechanism, we can achieve a significant improvement in terms of secrecy sum-rate and power efficiency.

On Second-Order Proto-Differentiability of Perturbation Maps

In this paper, second-order sensitivity analysis in vector optimization problems is considered. We prove that the efficient solution map and the efficient frontier map of a parameterized vector optimization problem are second-order proto-differentiable under some appropriate qualification conditions. Some sufficient conditions for inner and outer approximation of the second-order proto-derivative are also provided.

Asymptotic Farkas lemmas for convex systems

In this paper we establish characterizations of the containment of the set {xX: xC,g(x)K}{xX: f (x)0}, where C is a closed convex subset of a locally convex Hausdorff topological vector space, X, K is a closed convex cone in another locally convex Hausdorff topological vector space and g:X Y is a K- convex mapping, in a reverse convex set, define by the proper, lower semicontinuous, convex function. Here, no constraint qualification condition or qualification condition are assumed. The characterizations are often called asymptotic Farkas-type results. The second part of the paper was devoted to variant Asymptotic Farkas-type results where the mapping is a convex mapping with respect to an extended sublinear function. It is also shown that under some closedness conditions, these asymptotic Farkas lemmas go back to non-asymptotic Farkas lemmas or stable Farkas lemmas established recently in the literature. The results can be used to study the optimization

An approximate Hahn-Banach-Lagrange theorem

In this paper, we proved a new extended version of the Hahn-Banach-Lagrange theorem that is valid in the absence of a qualification condition and is called an approximate Hahn-Banach-Lagrange theorem. This result, in special cases, gives rise to approximate sandwich and approximate Hahn-Banach theorems. These results extend the Hahn-Banach-Lagrange theorem, the sandwich theorem in [18], and the celebrated Hahn-Banach theorem. The mentioned results extend the original ones into two features: Firstly, they extend the original versions to the case with extended sublinear functions (i.e., the sublinear functions that possibly possess extended real values). Secondly, they are topological versions which held without any qualification condition. Next, we showed that our approximate Hahn-Banach-Lagrange theorem was actually equivalent to the asymptotic Farkas-type results that were established recently [10]. This result, together with the results [5, 16], give us a general picture on the equivalence of the Farkas lemma and the Hahn-Banach theorem, from the original version to their corresponding extensions and in either non-asymptotic or asymptotic forms.

On Computation of Generalized Derivatives of the Normal-Cone Mapping and Their Applications

The paper concerns the computation of the graphical derivative and the regular (Fréchet) coderivative of the normal-cone mapping related to C2 inequality constraints under very weak qualification conditions. This enables us to provide the graphical derivative and the regular coderivative of the solution map to a class of parameterized generalized equations with the constraint set of the investigated type. On the basis of these results, we finally obtain a characterization of the isolated calmness property of the mentioned solution map and derive strong stationarity conditions for an MPEC with control constraints.

Sequential ε-Subdifferential Calculus for Scalar and Vector Mappings

In this paper, several sequential formulae are obtained for the Brondsted-Rockafellar e-subdifferential of the sum and the composition of two convex and lower semicontinuous mappings defined in reflexive Banach spaces. These calculus rules are stated in terms of limits of e-subgradients at nearby points to the nominal point without assuming any qualification condition, and extend some well-known sequential formulae for the Fenchel subdifferential. Next, as a consequence of these formulae, sequential sum and composition rules for a proper e-subdifferential of extended vector mappings are obtained, which involve a type of vector strong e-subdifferential.

The approach to the evaluation of options of the activities of the operators technical support information services telecommunication systems

The purpose of the article is the development of formalized models and algorithms for estimating the activity of operators working in telecommunication systems, specializing in providing IT services to the population and companies. The generalized structural method of the functional and structural theory of ergonomic and technical systems of professor A. I. Gubinsky is chosen as a basic solving method. The method is based on the use of standard functional units to describe the algorithms of activity. These units are presented in the form of functional networks. A number of formalized descriptions of algorithms of the operators’ activities for elimination of some problems of IT services providing is developed. The calculations of probability of timely and faultless execution of the activities algorithm for the elimination of one of possible problem situations are performed. The technology of the account at an assessment of reliability of activities for elimination of problems is shown. This technology considers such parameters as activity algorithms, operators’ qualification and working conditions at the workplaces. The advantage of the method consists in a possibility of receiving the unambiguous computer focused models and numerical estimation of various options of operators’ activities, who have different qualification. Using the proposed approach allows to develop the actions related to the search of ergonomic reserves of systems efficiency increase of providing access to resources of computer networks. The developed models can be the basis for decision support system for the optimization of operators’ activities. The difficulties of the method include the complexity of formation of initial data for the calculation and the necessity of supporting special statistical databases, also the need of systematic assessment of working conditions at the workplace of people working in the system

Majorization-Minimization Procedures and Convergence of SQP Methods for Semi-Algebraic and Tame Programs

In view of solving nonsmooth and nonconvex problems involving complex constraints (like standard NLP problems), we study general maximization-minimization procedures produced by families of strongly convex subproblems. Using techniques from semi-algebraic geometry and variational analysis—in particular Łojasiewicz inequality—we establish the convergence of sequences generated by these types of schemes to critical points. The broad applicability of this process is illustrated in the context of NLP. In that case, critical points coincide with KKT points. When the data are semi-algebraic or real analytic our method applies (for instance) to the study of various sequential quadratic programming (SQP) schemes: the moving balls method, the penalized SQP method and the extended SQP method. Under standard qualification conditions, this provides—to the best of our knowledge—the first general convergence results for general nonlinear programming problems. We emphasize the fact that, unlike most works on this subject, no second-order conditions and/or convexity assumptions whatsoever are made. Rate of convergence are shown to be of the same form as those commonly encountered with first-order methods.

Preservation of Prox-Regularity of Sets with Applications to Constrained Optimization

In this paper, we first provide counterexamples showing that sublevels of prox-regular functions and levels of differentiable mappings with Lipschitz derivatives may fail to be prox-regular. Then, we prove the uniform prox-regularity of such sets under usual verifiable qualification conditions. The preservation of uniform prox-regularity of intersection and inverse image under usual qualification conditions is also established. Applications to constrained optimization problems are given.

Towards Supremum-Sum Subdifferential Calculus Free of Qualification Conditions

We give a formula for the subdifferential of the sum of two convex functions where one of them is the supremum of an arbitrary family of convex functions. This is carried out under a weak assumption expressing a natural relationship between the lower semicontinuous envelopes of the data functions in the domain of the sum function. We also provide a new rule for the subdifferential of the sum of two convex functions, which uses a strategy of augmenting the involved functions. The main feature of our analysis is that no continuity-type condition is required. Our approach allows us to unify, recover, and extend different results in the recent literature.

Characterizing robust local error bounds for linear inequality systems under data uncertainty

We present necessary and sufficient conditions for the existence of robust local error bounds for linear inequality system in the face of data uncertainty where the uncertain data belong to a prescribed compact uncertainty set. The robust local error bound for an uncertain system is defined in terms of the existence of local error bound for its robust counterpart, where the uncertain linear inequality is enforced for every possible value of the data in the uncertainty set. Some of these conditions are expressed using the projection of the origin and others are given by way of normal cones at the boundary points of the solution set. In the case of commonly used interval data uncertainty, we show that a qualification condition completely characterizes the existence of robust local error bounds.

Boundedness of certain sets of Lagrange multipliers in vector optimization

In this paper, we establish Lagrange multiplier rules in terms of Michel–Penot subdifferential for nonsmooth vector optimization problem. A constraint qualification or regularity condition in terms of Michel–Penot subdifferential is given and under this regularity condition the boundedness of certain sets of Lagrange multipliers are discussed.

GENERALIZATIONS OF THE HAHN-BANACH THEOREM REVISITED

In this paper, based on the extended versions of the Farkas lemma for convex systems introduced recently in [9], we establish an extended version of a so called Hahn-Banach-Lagrange theorem introduced by Stephan Simons in [22]. This generalized version of the Hahn-Banach-Lagrange theorem holds in locally convex Hausdorff topological vector spaces under a Slater-type constraint qualification condition and with the relaxing of the lower semi-continuity of some functions involved and the closedness of the constrained sets. The version, in turn, yields extended versions of the Mazur-Orlicz theorem, the sandwich theorem, and the Hahn-Banach theorem concerning extended sublinear functions. It is then shown that all the generalized versions of the Farkas lemma for cone-convex/sublinear-convex systems in [9] and the new extended Hahn-Banach-Lagrange theorem just obtained are equivalent together. A class of composite problems involving sublinear-convex mappings is considered at the end of the paper. Here the main results of the paper are applied to get a strong duality result and optimality conditions for the class of problems. Moreover, a formula for the conjugate of the supremum of a family (possibly infinite, not lower semi-continuous) of convex functions is then derived from the duality result to show the generality and the significance of the class of problems in consideration.