Mangasarian-Fromovitz Constraint Qualification Research Articles

Degenerate Nonlinear Programming with a Quadratic Growth Condition

We show that the quadratic growth condition and the Mangasarian--Fromovitz constraint qualification (MFCQ) imply that local minima of nonlinear programs are isolated stationary points. As a result, when started sufficiently close to such points, an $L_\infty$ exact penalty sequential quadratic programming algorithm will induce at least R-linear convergence of the iterates to such a local minimum. We construct an example of a degenerate nonlinear program with a unique local minimum satisfying the quadratic growth and the MFCQ but for which no positive semidefinite augmented Lagrangian exists. We present numerical results obtained using several nonlinear programming packages on this example and discuss its implications for some algorithms.

On existence and uniqueness of stationary points in semi-infinite optimization

The paper deals with semi-infinite optimization problems which are defined by finitely many equality constraints and infinitely many inequality constraints. We generalize the concept of strongly stable stationary points which was introduced by Kojima for finite problems; it refers to the local existence and uniqueness of a stationary point for each sufficiently small perturbed problem, where perturbations up to second order are allowed. Under the extended Mangasarian-Fromovitz constraint qualification we present equivalent conditions for the strong stability of a considered stationary point in terms of first and second derivatives of the involved functions. In particular, we discuss the case where the reduction approach is not satisfied.

Modified Wilson's Method for Nonlinear Programs with Nonunique Multipliers

This paper first investigates Newton-type methods for generalized equations with compact solution sets. The analysis of their local convergence behavior is based, besides other conditions, on the upper Lipschitz-continuity of the local solution set mapping of a simply perturbed generalized equation. This approach is then applied to the KKT conditions of a nonlinear program with inequality constraints and leads to a modified version of the classical Wilson method. It is shown that the distances of the iterates to the set of KKT points converge q-quadratically to zero under conditions that do not imply a unique multiplier vector. Additionally to the Mangasarian-Fromovitz Constraint Qualification and to a Second-Order Sufficiency Condition the local minimizer is required to fulfill a Constant Rank Condition (weaker than the Constant Rank Constraint Qualification one of Janin) and a so-called Weak Complementarity Condition.

Stabilized Sequential Quadratic Programming

Recently, Wright proposed a stabilized sequential quadratic programming algorithm for inequality constrained optimization. Assuming the Mangasarian-Fromovitz constraint qualification and the existence of a strictly positive multiplier (but possibly dependent constraint gradients), he proved a local quadratic convergence result. In this paper, we establish quadratic convergence in cases where both strict complementarity and the Mangasarian-Fromovitz constraint qualification do not hold. The constraints on the stabilization parameter are relaxed, and linear convergence is demonstrated when the parameter is kept fixed. We show that the analysis of this method can be carried out using recent results for the stability of variational problems.

Vector optimization: Singularities, regularizations

We discuss three scalarizations of the multiobjectie optimization from the point of view of the parametric optimization. We analyze three important aspects: i) What kind of singularities may appear in the different parametrizations ii) Regularizations in the sense of Jongen, Jonker and Twilt, and in the sense of Kojima and Hirabayashi. iii) The Mangasarian-Fromovitz Constraint Qualification for the first parametrization.

The Crease Structure of the Karush-Kuhn-Tucker Set in Parametric Optimization

We study optimization problems depending on a parameter vector y. In particular, we consider the crease structure of the set Σ of pairs (x, y) where x is a Karush-Kuhn-Tucker point of the problem associated with parameter y; the Mangasarian-Fromovitz constraint qualification is assumed to hold. We present a lower bound for the fineness of Whitney regular stratifications of Σ. This provides an approximation of its crease structure.

On Uniqueness of Lagrange Multipliers in Composite Optimization

In this paper, we introduce a new constraint qualification for composite optimization problems. We show that this constraint qualification is a necessary and sufficient condition for the uniqueness of Lagrange multipliers. We also show that it is a natural extension of the strict Mangasarian–Fromovitz constraint qualification to the composite optimization setting.

Dini Derivatives of the Marginal Function of a Non-Lipschitzian Program

Upper and lower bounds are established for the Dini directional derivatives of the marginal function of a parametric mathematical program. In this program, the equality constraint functions are assumed to be strictly differentiable, but the objective and inequality constraint functions can belong to a large class of non-Lipschitzian functions. A nonsmooth version of the Mangasarian–Fromovitz constraint qualification is also assumed. The main tool in the proofs of these bounds is the calculus of tangent cones.

On the Topology of the Karush–Kuhn–Tucker Set under Mangasarian–Fromovitz Constraint Qualification

This paper deals with smooth optimization problems $\cal p$ in $\R^n$ depending on parameter $y\in\R^p$. The problem ${\cal p} (y)$ is defined by means of a finite number of equality and inequality constraints. We study the set ${\bf \sum}_{KKT}$ of pairs $(x,y)$ such that $x$ is a Karush-- Kuhn--Tucker point of the problem ${\cal p} (y)$. Let ${\bf \sum}$ denote the subset of ${\bf \sum}_{KKT}$ at which the Mangasarian--Fromovitz constraint qualification is fulfilled. For problem data in general position we prove that ${\bf \sum}$ is a topological manifold of dimension $p$.

On equivalent stability properties in semi-infinite optimization

This paper studies noncompact feasible sets of a semi-infinite optimization problem which are defined by finitely many equality constraints and infinitely many inequality constraints. The main result is the equivalence of the overall validity of the Extended Mangasarian Fromovitz Constraint Qualification with certain (topological) stability conditions. Furthermore, two perturbation theorems being of independent interest are presented.

Strong Stability in Variational Inequalities

In this paper we consider a generalization of Kojima's strong stability in nonlinear programs to variational inequalities constrained by a system of equations and inequalities. Roughly speaking, strong stability refers to the local existence and uniqueness of a solution of a system under small perturbations. The purpose of the paper is to establish a new and complete characterization for strongly stable generalized Karush-Kuhn-Tucker points and to give a complete characterization for strongly stable stationary solutions under the Mangasarian-Fromovitz constraint qualification.

One-Parametric Semi-Infinite Optimization: On the Stability of the Feasible Set

This paper studies a global stability property of the (noncompact) feasible set $M( H,G,t )$ of a semi-infinite optimization problem defined by finitely many equations $H( x,t ) = 0$ and, perhaps, infinitely many inequalities $G( x,t,y ) \leq 0$ that depend on a real parameter t that varies in a compact parameter interval T. Global stability refers to the homeomorphy of $M[ H,G,t_1 ]$ and $M[ H,G,t_2 ]$ for any parameter values $t_1 ,t_2 \in T$. It is shown that the overall validity of a so-called extended Mangasarian–Fromovitz constraint qualification at infinity (a constraint qualification taking parameter information into account) is sufficient for the stability mentioned.

A nonsmooth variant of the Mangasarian-Fromovitz constraint qualification

A unified view on constraint qualifications for nonsmooth equality and inequality constrained programs is presented. A fairly general constraint qualification for programs involving B-differential functions is given. Further specification to piecewise differentiable equality constraints and locally Lipschitz continuous inequality constraints yields a nonsmooth version of the Mangasarian-Fromovitz constraint qualification.

Metric regularity of the feasible set mapping in semi-infinite optimization

For parametric systems of (finitely many) equations and (infinitely many) inequalities the well-known concept of metric regularity is shown to be equivalent to the so-called extended Mangasarian-Fromovitz constraint qualification. By this, a corresponding result obtained by Robinson for finite optimization problems my be transferred to semi-infinite optimization. For the proof a local epigraph representation of the constraint set is mainly used.

Manifold Structure of the Karush–Kuhn–Tucker Stationary Solution Set with Two Parameters

This paper deals with a 2-dimensional parameter family of nonlinear programs: minimize $h_0 ( x,t )$ subject to the equality constraints $h_i (x,t) = 0\, (i = 1,\ldots ,l)$ and the inequality constraints $h_j ( x,t ) \leq 0\,( j = l + 1, \ldots ,m )$. Each $h_i \, ( i = 0,1, \ldots ,m )$ is a twice continuously differentiable real-valued map defined on the $( n + 2 )$-dimensional Euclidean space $R^{n + 2} $, where $x \in R^n $ denotes a variable vector and $t \in R^2 $ denotes a 2-dimensional parameter vector. The local properties of the Karush–Kuhn–Tucker stationary solution set, the set $\Sigma $ consisting of all $( x,t )$ such that x is a stationary solution of the program for some t, are studied. In fact, it is shown that if the Mangasarian-Fromovitz constraint qualification and a regular value condition are satisfied, (i) the set $\Sigma $ is a 2-dimensional topological manifold without a boundary, and (ii) the set $\Sigma $ is a generalized creased manifold if, in addition, a constant rank condition holds.

Semi-infinite optimization: Structure and stability of the feasible set

The problem of the minimization of a functionf: ℝn→ℝ under finitely many equality constraints and perhaps infinitely many inequality constraints gives rise to a structural analysis of the feasible setM[H, G]={x∈ℝn¦H(x)=0,G(x, y)≥0,y∈Y} with compactY⊂ℝr. An extension of the well-known Mangasarian-Fromovitz constraint qualification (EMFCQ) is introduced. The main result for compactM[H, G] is the equivalence of the topological stability of the feasible setM[H, G] and the validity of EMFCQ. As a byproduct, we obtain under EMFCQ that the feasible set admits local linearizations and also thatM[H, G] depends continuously on the pair (H, G). Moreover, EMFCQ is shown to be satisfied generically.

On constraint qualifications

The linear independence constraint qualification (LICQ) and the weaker Mangasarian-Fromovitz constraint qualification (MFCQ) are well-known concepts in nonlinear optimization. A theorem is proved suggesting that the set of feasible points for which MFCQ essentially differs from LICQ is small in a specified sense. As an auxiliary result, it is shown that, under MFCQ, the constraint set (even in semi-infinite optimization) is locally representable in epigraph form.

An optimality test for semi-infinite linear programming

In this paper we present a test to characterize the optimal solutions for the continuous semi-infinite linear programming problem. This optimality characterization is a condition of Kuhn–Tucker type. The resolution of a linear program permits to check the optimality of a feasible point,to detect the unboundedness of the problem and to find descent directions. We give some illustrative examples. We show that the local Mangasarian–Fromovitz constraint qualification is almost equivalent to Slater qualification for this problem. Furthermore, it follows from our study that this optimality condition is always necessary for a wide class of semi-infinite linear programming problems

Sensitivity Analysis of Nonlinear Programs and Differentiability Properties of Metric Projections

This paper is concerned with a study of differentiability properties of the optimal value function and an associated optimal solution of a parametrized nonlinear program. Second order analysis is presented essentially under the Mangasarian-Fromovitz constraint qualification when the corresponding vector of Lagrange multipliers is not necessarily unique. It is shown that under certain regularity conditions the optimal value function possesses second order directional derivatives and the optimal solution mapping is directionally differentiable. The results obtained are applied to an investigation of metric projections in finite-dimensional spaces.

Structural stability in nonlinear optimization

This paper is concerned with global stability properties for differentiate optimization problems of the type: We introduce a natural equivalence concept for optimization problems, In this equivalence each lower level set of one problem is mapped homeomorphically onto a corresponding lower level set of the other one. In case that 𝔓([fbar],ℏ,g) is equivalent with 𝔓(f,H,G) for all ([fbar],ℏ,g) in some neighborhood of (f. H. G) we call 𝔓(f, H, G) structurally stable; the topology used takes derivatives up to order two into account. Under the assumption that M[H, G] is compact we prove that structural stability of 𝔓(f,H,G) implies the following three conditions: C1. The Mangasarian-Fromovitz constraint qualification is satisfied at every point of M[H, G]. C2. Every Kuhn-Tucker point of 𝔓(f,H,G) is strongly stable in the sense of Kojima. C3. Different Kuhn-Tucker points have different (f-) val es. On the other hand, again in case M[H,G] is compact, we conjecture that the conditions CI, C2 and C3 imply structural stability.