Mangasarian-Fromovitz Constraint Qualification Research Articles

On the Mangasarian–Fromovitz constraint qualification and Karush–Kuhn–Tucker conditions in nonsmooth semi-infinite multiobjective programming

We discuss constraint qualifications in Karush–Kuhn–Tucker multiplier rules in nonsmooth semi-infinite multiobjective programming. A version of the Manganarian–Fromovitz constraint qualification is proposed, in terms of the Michel–Penot directional derivative and the Studniarski derivative of order p which is just the order of the directional Holder metric subregularity which is included also in this proposed qualification version. Using this qualification together with the Pshenichnyi–Levitin–Valadire property, we establish Karush–Kuhn–Tucker optimality conditions for Borwein-proper and firm solutions. We also compare in detail our qualification version with other usually-employed constraint qualifications. Applications to semi-infinite multiobjective fractional problems and minimax problems are provided.

Convergence properties of a class of exact penalty methods for semi-infinite optimization problems

In this paper, a new class of unified penalty functions are derived for the semi-infinite optimization problems, which include many penalty functions as special cases. They are proved to be exact in the sense that under Mangasarian–Fromovitz constraint qualification conditions, a local solution of penalty problem is a corresponding local solution of original problem when the penalty parameter is sufficiently large. Furthermore, global convergence properties are shown under some conditions. The paper is concluded with some numerical examples proving the applicability of our methods to PID controller design and linear-phase FIR digital filter design.

A class of exact penalty functions and penalty algorithms for nonsmooth constrained optimization problems

In this paper, a class of smoothing penalty functions is proposed for optimization problems with equality, inequality and bound constraints. It is proved exact, under the condition of weakly generalized Mangasarian–Fromovitz constraint qualification, in the sense that each local optimizer of the penalty function corresponds to a local optimizer of the original problem. Furthermore, necessary and sufficient conditions are discussed for the inverse proposition of exact penalization. Based on the theoretical results in this paper, a class of smoothing penalty algorithms with feasibility verification is presented. Theories on the penalty exactness, feasibility verification and global convergence of the proposed algorithm are presented. Numerical results show that this algorithm is effective for nonsmooth nonconvex constrained optimization problems.

Necessary conditions for continuous-time optimization under the Mangasarian–Fromovitz constraint qualification

ABSTRACTIn this paper, a Mangasarian–Fromovitz-type regularity condition is presented with the purpose of obtaining first order necessary optimality conditions for continuous-time nonlinear programming problems with both equality and inequality constraints. The Karush–Kuhn–Tucker-type optimality conditions are obtained by means of geometric ideas, where a theorem of the alternative and a uniform implicit function theorem are the main tools.

Wolfe-Type Duality for Mathematical Programs with Equilibrium Constraints

This paper considers the mathematical programs with equilibrium constraints (MPEC). It is well-known that, due to the existence of equilibrium constraints, the Mangasarian-Fromovitz constraint qualification does not hold at any feasible point of MPEC and hence, in general, the developed numerical algorithms for standard nonlinear programming problems can not be applied to solve MPEC directly. During the past two decades, much research has been done to develop numerical algorithms and study optimality, stability, and sensitivity for MPEC. However, there are very few results on duality for MPEC in the literature. In this paper, we present a Wolfe-type duality for MPEC and, under some suitable conditions, we establish various duality theorems such as the weak duality, direct duality, converse duality, and strict converse duality theorems. We further show that a linear MPEC is equivalent to a linear programming problem in some sense.

Second-order optimality conditions and Lagrange multiplier characterizations of the solution set in quasiconvex programming

ABSTRACTSecond-order optimality conditions for the vector nonlinear programming problems with inequality constraints and continuously differentiable data are studied in this paper. We introduce a new second-order constraint qualification, which includes Mangasarian-Fromovitz constraint qualification as a particular case. We obtain necessary and sufficient conditions for weak efficiency of problems with a second-order pseudoconvex vector objective function and quasiconvex constraints. We also derive Lagrange multiplier characterizations of the solution set of a scalar problem with a second-order pseudoconvex objective function and quasiconvex inequality constraints, provided that one of the solutions and the Lagrange multipliers in the Karush-Kuhn-Tucker conditions are known. At last, we introduce notions of a second-order pseudoconvex vector function and KKT-pseudoconvex vector problem with inequality constraints. We derive necessary and sufficient conditions for efficiency in the vector problem with inequality constraints. Three examples are presented.

Necessary optimality conditions for a bilevel multiobjective programming problem via a Ψ-reformulation

ABSTRACTIn this paper, we are concerned with a bilevel multiobjective optimization problem . First, using Ψ, a function introduced by Gadhi and Dempe [Necessary optimality conditions and a new approach tomulti-objective bilevel optimization problems. J Optim Theory Appl. 2012;155:100–114], we transform into a one level optimization problem . Second, on terms of convexificators, using a scalarization technique, we derive a Karash-Kuhn-Tucker (KKT)-type necessary optimality conditions to the initial problem under a generalized Abadie constraint qualification without the assumption that the lower-level problem satisfies the Mangasarian Fromovitz constraint qualification. Some examples have been introduced to illustrate our results.

Uniqueness of multipliers in optimal control: the missing piece

Abstract In a well-known paper by Kyparisis it is proved that, in nonlinear programming, the uniqueness of Lagrange multipliers is equivalent to a strict version of the Mangasarian–Fromovitz constraint qualification which, in turn, implies the satisfaction of second-order necessary optimality conditions. This is no longer the case in optimal control where, as shown in a recent paper, the corresponding strict constraint qualification is only sufficient for the uniqueness of multipliers. In this paper we exhibit the missing piece: a new, simple condition, implied by the strict constraint qualification, which is necessary and sufficient for the uniqueness of multipliers in optimal control.

A New Smoothing Method for Mathematical Programs with Complementarity Constraints Based on Logarithm-Exponential Function

We present a new smoothing method based on a logarithm-exponential function for mathematical program with complementarity constraints (MPCC). Different from the existing smoothing methods available in the literature, we construct an approximate smooth problem of MPCC by partly smoothing the complementarity constraints. With this new method, it is proved that the Mangasarian-Fromovitz constraint qualification holds for the approximate smooth problem. Convergence of the approximate solution sequence, generated by solving a series of smooth perturbed subproblems, is investigated. Under the weaker constraint qualification MPCC-Cone-Continuity Property, it is proved that any accumulation point of the approximate solution sequence is a M-stationary point of the original MPCC. Preliminary numerical results indicate that the developed algorithm based on the partly smoothing method is efficient, particularly in comparison with the other similar ones.

On a Conjecture in Second-Order Optimality Conditions

In this paper, we deal with a conjecture formulated in Andreani et al. (Optimization 56:529–542, 2007), which states that whenever a local minimizer of a nonlinear optimization problem fulfills the Mangasarian–Fromovitz constraint qualification and the rank of the set of gradients of active constraints increases at most by one in a neighborhood of the minimizer, a second-order optimality condition that depends on one single Lagrange multiplier is satisfied. This conjecture generalizes previous results under a constant rank assumption or under a rank deficiency of at most one. We prove the conjecture under the additional assumption that the Jacobian matrix has a smooth singular value decomposition. Our proof also extends to the case of the strong second-order condition, defined in terms of the critical cone instead of the critical subspace.

Fast Sensitivity-Based Nonlinear Economic Model Predictive Control with Degenerate NLP

We present a fast sensitivity-based nonlinear model predictive control (NMPC) algorithm, that can handle non-unique multipliers in the discretized dynamic optimization problem. Non-unique multipliers may arise, for example when path constraints are active for longer periods of the prediction horizon. This is a common situation in economic model predictive control. In such cases, the optimal nonlinear programming (NLP) solution often satisfies the Mangasarian-Fromovitz constraint qualification (MFCQ), which implies non-unique, but bounded multipliers. Consequently, any sensitivity-based fast NMPC scheme must allow for discontinuous jumps in the multipliers. In this paper, we apply a sensitivity-based path-following algorithm that allows multiplier jumps within the advance-step NMPC (asNMPC) framework. The path-following method consists of a corrector and a predictor step, which are computed by solving a system of linear equations, and a quadratic programming problem, respectively, and a multiplier jump step determined by the solution of a linear program. We demonstrate the proposed method on an economic NMPC case study with a CSTR.

Normality and uniqueness of Lagrange multipliers

In this paper we study, for certain problems in the calculus of variations and optimal control, two different questions related to uniqueness of multipliers appearing in first order necessary conditions. One deals with conditions under which a given multiplier associated with an extremal of a fixed function is unique, a property which, in nonlinear programming, is known to be equivalent to the strict Mangasarian-Fromovitz constraint qualification. We show that, for isoperimetric problems in the calculus of variations, a similar characterization holds, but not in optimal control where the corresponding condition is only sufficient for the uniqueness of the multiplier. The other question is related to the set of multipliers associated with all functions for which a solution to the constrained problem is given. We prove that, for both types of problems, this set is a singleton if and only if a strong normality assumption holds.

Some Results about the Isolated Calmness of a Mixed Variational Inequality Problem

It is well known that optimization problem model has many applications arising from matrix completion, image processing, statistical learning, economics, engineering sciences, and so on. And convex programming problem is closely related to variational inequality problem. The so-called alternative direction of multiplier method (ADMM) is an importance class of numerical methods for solving convex programming problem. When analyzing the rate of convergence of various ADMMs, an error bound condition is usually required. The error bound can be obtained when the isolated calmness of the inverse of the KKT mapping of the related problem holds at the given KKT point. This paper is to study the isolated calmness of the inverse KKT mapping onto the mixed variational inequality problem with nonlinear term defined by norm function and indicator function of a convex polyhedral set, respectively. We also consider the isolated calmness of the inverse KKT mapping onto classical variational inequality problem with equality and inequality constrains under strict Mangasarian-Fromovitz constraint qualification condition. The results obtained here are new and very interesting.

Error bounds for affine variational inequalities with second-order cone constraints

In this paper, error bounds for affine variational inequalities with second-order cone constraints are considered. Examples are given to show that, in general, Lipschitz error bounds may be invalid for affine second-order cone inclusion problems. We provide a sufficient condition (not stronger than Mangasarian–Fromovitz constraint qualification), under which a local Lipschitz error bound is valid for the variational inequality problem. Moreover, under a full row rank assumption, a local Hölder error bound is established for the variational inequality problem and the Hölder exponent is bounded by a function of problem dimensions.

Convexificators and boundedness of the Kuhn–Tucker multipliers set

In this paper we consider a nonsmooth optimization problem with equality, inequality and set constraints. We propose new constraint qualifications and Kuhn–Tucker type necessary optimality conditions for this problem involving locally Lipschitz functions. The main tool of our approach is the notion of convexificators. We introduce a nonsmooth version of the Mangasarian–Fromovitz constraint qualification and show that this constraint qualification is necessary and sufficient for the Kuhn–Tucker multipliers set to be nonempty and bounded.

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.

A smoothing augmented Lagrangian method for nonconvex, nonsmooth constrained programs and its applications to bilevel problems

In this paper, we consider a class of nonsmooth and nonconvex optimization problem with an abstract constraint. We propose an augmented Lagrangian method for solving the problem and construct global convergence under a weakly nonsmooth Mangasarian-Fromovitz constraint qualification. We show that any accumulation point of the iteration sequence generated by the algorithm is a feasible point which satisfies the first order necessary optimality condition provided that the penalty parameters are bounded and the upper bound of the augmented Lagrangian functions along the approximated solution sequence exists. Numerical experiments show that the algorithm is efficient for obtaining stationary points of general nonsmooth and nonconvex optimization problems, including the bilevel program which will never satisfy the nonsmooth Mangasarian-Fromovitz constraint qualification.

On second-order optimality conditions in nonlinear optimization

In this work we present new weak conditions that ensure the validity of necessary second-order optimality conditions (SOC) for nonlinear optimization. We are able to prove that weak and strong SOCs hold for all Lagrange multipliers using Abadie-type assumptions. We also prove weak and strong SOCs for at least one Lagrange multiplier imposing the Mangasarian–Fromovitz constraint qualification and a weak constant rank assumption.

An Introduction to Vector Variational Inequalities and Some New Results

This survey paper discusses some new results on vector variational inequalities. It can serve as an elementary introduction to vector variational inequalities and vector optimization problems. The focus point is made on the results about connectedness structure of the solution sets, which are obtained by a scalarization method and properties of semi-algebraic sets. The first major theorem says that both Pareto solution set and weak Pareto solution set of a vector variational inequality, where the constraint set is polyhedral convex and the basic operators are given by polynomial functions, have finitely many connected components. The second major theorem asserts that both proper Pareto solution set and weak Pareto solution set of a vector variational inequality, where the convex constraint set is given by polynomial functions and all the components of the basic operators are polynomial functions, have finitely many connected components, provided that the Mangasarian-Fromovitz Constraint Qualification is satisfied at every point of the constraint set. In addition, it has been established that if the proper Pareto solution set is dense in the Pareto solution set, then the latter also has finitely many connected components. Consequences of the results for vector optimization problems are shown.

A nonsmooth approach to envelope theorems

We develop a nonsmooth approach to envelope theorems applicable to a broad class of parameterized constrained nonlinear optimization problems that arise typically in economic applications with nonconvexities and/or nonsmooth objectives. Our methods emphasize the role of the Strict Mangasarian–Fromovitz Constraint Qualification (SMFCQ), and include envelope theorems for both the convex and nonconvex case, allow for noninterior solutions as well as equality and inequality constraints. We give new sufficient conditions for the value function to be directionally differentiable, as well as continuously differentiable. We apply our results to stochastic growth models with Markov shocks and constrained lattice programming problems.