Farkas Minkowski Research Articles

Around a Farkas type lemma

The first two authors of this paper asserted in Lemma 4 of ‘New Farkas-type constraint qualifications in convex infinite programming’ (DOI: 10.1051/cocv:2007027) that a given reverse convex inequality is consequence of a given convex system satisfying the Farkas–Minkowski constraint qualification if and only if certain set depending on the data contains a particular point of the vertical axis. This paper identifies a hidden assumption in this reverse Farkas lemma which always holds in its applications to nontrivial optimization problems. Moreover, it shows that the statement remains valid when the Farkas–Minkowski constraint qualification fails by replacing the mentioned set by its closure. This hidden assumption is also characterized in terms of the data. Finally, the paper provides some applications to convex infinite systems and to convex infinite optimization problems.

Lagrangian multipliers for generalized affine and generalized convex vector optimization problems of set-valued maps

Abstract In this paper, we introduce some definitions of generalized affine set-valued maps: affinelike, preaffinelike, nearaffinelike, and prenearaffinelike maps. We present examples to explain that our definitions of generalized affine maps are different from each other. We derive a theorem of alternative of Farkas–Minkowski type, discuss Lagrangian multipliers for constrained set-valued optimization problems, and obtain some optimality conditions for weakly efficient solutions.

ϵ-Efficient solutions in semi-infinite multiobjective optimization

In this paper we apply some tools of nonsmooth analysis and scalarization method due to Chankong–Haimes to find ϵ-efficient solutions of semi-infinite multiobjective optimization problems (MP). We establish ϵ-optimality conditions of Karush–Kuhn–Tucker (KKT) type under Farkas–Minkowski (FM) constraint qualification by using ϵ-subdifferential concept. In addition we propose mixed type dual problem (including dual problems of Wolfe and Mond–Weir types as special cases) for ϵ-efficient solutions and investigate relationship between mentioned (MP) and its dual problem as well as establish several ϵ-duality theorems.

A proof of the Farkas–Minkowski theorem by a tandem method

AbstractThis note presents a proof of the Farkas–Minkowski theorem. Our proof does not presuppose the closedness of a finitely generated cone, nor employs separation theorems either. Even the concept of linear independence or invertibility of matrices is not necessary. Our new device consists in proving the Farkas–Minkowski theorem and the closedness of a finitely generated cone at the same time based upon mathematical induction. We make use of a minimization problem with an equality constraint, a method familiar to economics students.

Constraint qualifications and optimality conditions for nonconvex semi-infinite and infinite programs

The paper concerns the study of new classes of nonlinear and nonconvex optimization problems of the so-called infinite programming that are generally defined on infinite-dimensional spaces of decision variables and contain infinitely many of equality and inequality constraints with arbitrary (may not be compact) index sets. These problems reduce to semi-infinite programs in the case of finite-dimensional spaces of decision variables. We extend the classical Mangasarian–Fromovitz and Farkas–Minkowski constraint qualifications to such infinite and semi-infinite programs. The new qualification conditions are used for exact calculations of the appropriate normal cones to sets of feasible solutions for these programs by employing advanced tools of variational analysis and generalized differentiation. In the further development we derive first-order necessary optimality conditions for infinite and semi-infinite programs, which are new in both finite-dimensional and infinite-dimensional settings.

Quasi-Slater and Farkas--Minkowski Qualifications for Semi-infinite Programming with Applications

The well-known Farkas--Minkowski (FM) type qualification plays an important role in linear semi-infinite programming and has been extensively developed by many authors in establishing optimality conditions, duality, and stability for semi-infinite programming. In this paper, we introduce the concept of the quasi-Slater condition for a semi-infinite convex inequality system and present that the Slater type conditions imply the FM qualification under some appropriate continuity assumption of the set-valued mapping $i\mapsto f_i(x)$. Applying these relationships, we establish dual characterizations, both asymptotic and nonasymptotic, for set containment problems and provide some sufficient conditions for ensuring the strong Lagrangian duality and Farkas lemma.

Subdifferentials of Marginal Functions in Semi-infinite Programming

This paper proposes two new constraint qualification conditions (CQs) which are useful for a unified study of CQs from both a convex analysis and a nonsmooth analysis point of view. Our CQs cover the existing CQs of Mangasarian–Fromovitz and Farkas–Minkowski types. Some sufficient conditions for the validity of the new CQs are given. Under these CQs, we derive formulae for computing and/or estimating the (basic and singular) subdifferentials of marginal/optimal value function in semi-infinite programming from some results of modern variational analysis and generalized differentiation. Examples are given to illustrate the obtained formulae.

Linear representations and quasipolyhedrality of a finite-valued convex function†

The possibility of representing the epigraph of a finite-valued convex function by means of a (locally) Farkas–Minkowski linear semi-infinite inequalities system is studied in this article. Moreover, we prove that the so-called locally polyhedral representations characterize the function, giving rise to the concept of the quasipolyhedral function. Conditions for its conjugate to be also quasipolyhedral are obtained, as well as the characterization of its subdifferential and ϵ-subdifferential in terms of a specific sequence of ordinary polyhedral functions. †Dedicated to H. Th. Jongen on the occasion of his 60th birthday.

New regularity conditions for strong and total Fenchel–Lagrange duality in infinite dimensional spaces

We give new regularity conditions for convex optimization problems in separated locally convex spaces. We completely characterize the stable strong and strong Fenchel–Lagrange duality. Then we give similar statements for the case when a solution of the primal problem is assumed as known, obtaining complete characterizations for the so-called total and stable total Fenchel–Lagrange duality, respectively. For particular settings the conditions that we consider turn into some constraint qualifications already used by different authors, like Farkas–Minkowski CQ , locally Farkas–Minkowski CQ and basic CQ , and we rediscover and improve some recent results from the literature.

On strong and total Lagrange duality for convex optimization problems

We give some necessary and sufficient conditions which completely characterize the strong and total Lagrange duality, respectively, for convex optimization problems in separated locally convex spaces. We also prove similar statements for the problems obtained by perturbing the objective functions of the primal problems by arbitrary linear functionals. In the particular case when we deal with convex optimization problems having infinitely many convex inequalities as constraints the conditions we work with turn into the so-called Farkas–Minkowski and locally Farkas–Minkowski conditions for systems of convex inequalities, recently used in the literature. Moreover, we show that our new results extend some existing ones in the literature.

Necessary and sufficient constraint qualifications for solvability of systems of infinite convex inequalities

In this paper we present constraint qualifications which completely characterize the Farkas–Minkowski and the locally Farkas–Minkowski convex (possibly infinite) inequality systems posed in topological vector spaces. The number of constraints and the dimension of the linear space are arbitrary (possibly infinite). The constraint qualifications considered in this paper are expressed in terms of the solvability of certain parametric convex (linear) systems and the uniform strong duality or the uniform min–max duality relative to the Lagrange (Haar) dual problems of suitable convex (linear) parametric optimization problems.

Polyhedral cones and monomial blowing-ups

We show that a polyhedral cone Γ in R n with apex at 0 can be brought to the first quadrant by a finite sequence of monomial blowing-ups if and only if Γ ∩ ( - R ⩾ n ) = { 0 } . The proof is non-trivially derived from the theorem of Farkas–Minkowski. Then, we apply this theorem to show how the Newton diagrams of the roots of any Weierstraß polynomial P ( x , z ) = z m + h 1 ( x ) z m - 1 + ⋯ + h m - 1 ( x ) z + h m ( x ) , h i ( x ) ∈ k 〚 x 1 , … , x n 〛 [ z ] , are contained in a polyhedral cone of this type.

Locally Farkas–Minkowski Systems in Convex Semi-Infinite Programming

A pair of constraint qualifications in convex semi-infinite programming, namely the locally Farkas–Minkowski constraint qualification and generalized Slater constraint qualification, are studied in the paper. We analyze the relationship between them, as well as the behavior of the so-called active and sup-active mappings, accounting for the tightness of the constraint system at each point of the variables space. The generalized Slater constraint qualification guarantees a regular behavior of the supremum function (defined as supremum of the infinitely many functions involved in the constraint system), giving rise to the well-known Valadier formula.

Lagrangian Duality for Preinvex Set-Valued Functions

In this paper, generalizing the concept of cone convexity, we have defined cone preinvexity for set-valued functions and given an example in support of this generalization. A Farkas–Minkowski type theorem has been proved for these functions. A Lagrangian type dual has been defined for a fractional programming problem involving preinvex set-valued functions and duality results are established.

Dimension and finite reduction in linear semi-infinite programming

In the first part, different characterizations for the dimension of the feasible set in linear semi-infinite programming are provided. They involve the corresponding dimensions of some parameter sets, as the consequent inequalities cone and its lineality subspace. The remaining sections of the paper deal with Farkas–Minkowski systems. The third section is devoted to establish some results concerning the optimal set and its dimension, exploiting its strong relation with a particular parameter cone associated with the corresponding unstable constraints. The last section approaches the finite reducibility problem. We have intended to characterize those finite subproblems with the same optimal value as the original problem, by means of a simplc dual analysis, based on the main results derived before.