Mangasarian-Fromovitz Constraint Qualification Research Articles

A trust-region scheme for constrained multi-objective optimization problems with superlinear convergence property

In this paper, a numerical approximation method is developed to find approximate solutions to a class of constrained multi-objective optimization problems. All the functions of the problem are not necessarily convex functions. At each iteration of the method, a particular type of subproblem is solved using the trust region technique, and the step is evaluated using the notions of actual reduction and predicted reduction. A non-differentiable l ∞ penalty function restricts the constraint violations. An adaptive BFGS update formula is introduced. Global convergence of the proposed algorithm is established under the Mangasarian-Fromovitz constraint qualification and some mild assumptions. Furthermore, it is justified that the proposed algorithm displays a super-linear convergence rate. Numerical results are provided to show the efficiency of the algorithm in the quality of the approximated Pareto front.

A constraint qualification for the dislocation hyperbolic augmented Lagrangian algorithm

In this paper, we study an augmented Lagrangian-type algorithm called the Dislocation Hyperbolic Augmented Lagrangian Algorithm (DHALA), which solves an inequality nonconvex optimization problem. We show that the sequence generated by DHALA converges to a Karush–Kuhn–Tucker (KKT) point under the Mangasarian–Fromovitz constraint qualification. The contribution of our work is to consider a constraint qualification into this algorithm. Finally, we present some computational illustrations to demonstrate the performance our algorithm works.

On the weakest constraint qualification for sharp local minimizers

The sharp local minimality of feasible points of nonlinear optimization problems is known to possess a characterization by a strengthened version of the Karush–Kuhn–Tucker conditions, as long as the Mangasarian–Fromovitz constraint qualification holds. This strengthened condition is not easy to check algorithmically since it involves the topological interior of some set. In this article, we derive an algorithmically tractable version of this condition, called strong Karush–Kuhn–Tucker condition. We show that the Guignard constraint qualification is the weakest condition under which a feasible point is a strong Karush–Kuhn–Tucker point for every continuously differentiable objective function possessing the point as a sharp local minimizer. As an application, our results yield an algebraic characterization of strict local minimizers of linear programs with cardinality constraints.

Solving Bilevel Programs Based on Lower-Level Mond-Weir Duality

This paper focuses on developing effective algorithms for solving a bilevel program. The most popular approach is to replace the lower-level problem with its Karush-Kuhn-Tucker conditions to generate a mathematical program with complementarity constraints (MPCC). However, MPCC does not satisfy the Mangasarian-Fromovitz constraint qualification (MFCQ) at any feasible point. In this paper, inspired by a recent work using the lower-level Wolfe duality (WDP), we apply the lower-level Mond-Weir duality to present a new reformulation, called MDP, for bilevel program. It is shown that, under mild assumptions, they are equivalent in globally or locally optimal sense. An example is given to show that, different from MPCC, MDP may satisfy the MFCQ at its feasible points. Relations among MDP, WDP, and MPCC are investigated. On this basis, we extend the MDP reformulation to present another new reformulation (called eMDP), which has similar properties to MDP. Furthermore, to compare two new reformulations with the MPCC and WDP approaches, we design a procedure to generate 150 tested problems randomly and comprehensive numerical experiments show that MDP has quite evident advantages over MPCC and WDP in terms of feasibility to the original bilevel programs, success efficiency, and average CPU time, whereas eMDP is far superior to all other three reformulations. History: Accepted by Pascal Van Hentenryck, Area Editor for Computational Modeling: Methods & Analysis. Funding: This work was supported by the National Natural Science Foundation of China [Grants 12071280 and 11901380]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.0108 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0108 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Nonregular multiobjective problems: optimality conditions

Necessary and sufficient conditions for multiobjective problems with both equality constraints and inequality constraints are presented. These conditions are applied when the constraints of the problem do not necessarily satisfy any regularity assumptions and they are based on the concept of p-regularity, introduced by Tret'yakov (1983, Necessary conditions for optimality of pth order. Control Optim.;28–35). We introduce the notion of generalized Mangasarian-Fromovitz constraint qualification. This new concept is used to obtain our results.

Bilevel hyperparameter optimization for support vector classification: theoretical analysis and a solution method

Support vector classification (SVC) is a classical and well-performed learning method for classification problems. A regularization parameter, which significantly affects the classification performance, has to be chosen and this is usually done by the cross-validation procedure. In this paper, we reformulate the hyperparameter selection problem for support vector classification as a bilevel optimization problem in which the upper-level problem minimizes the average number of misclassified data points over all the cross-validation folds, and the lower-level problems are the l_1-loss SVC problems, with each one for each fold in T-fold cross-validation. The resulting bilevel optimization model is then converted to a mathematical program with equilibrium constraints (MPEC). To solve this MPEC, we propose a global relaxation cross-validation algorithm (GR–CV) based on the well-know Sholtes-type global relaxation method (GRM). It is proven to converge to a C-stationary point. Moreover, we prove that the MPEC-tailored version of the Mangasarian–Fromovitz constraint qualification (MFCQ), which is a key property to guarantee the convergence of the GRM, automatically holds at each feasible point of this MPEC. Extensive numerical results verify the efficiency of the proposed approach. In particular, compared with other methods, our algorithm enjoys superior generalization performance over almost all the data sets used in this paper.

Existence of Pareto Solutions for Vector Polynomial Optimization Problems with Constraints

This paper deals with a vector polynomial optimization problem over a basic closed semi-algebraic set. By invoking some powerful tools from real semi-algebraic geometry, we first introduce the concept called tangency varieties; obtain the relationships of the Palais–Smale condition, Cerami condition, M-tameness, and properness related to the considered problem, in which the condition of Mangasarian–Fromovitz constraint qualification at infinity plays an essential role in deriving these relationships. At last, according to the obtained connections, we establish the existence of Pareto solutions to the problem in consideration and give some examples to illustrate our main findings.

Running Primal-Dual Gradient Method for Time-Varying Nonconvex Problems

This paper considers a nonconvex optimization problem that evolves over time, and addresses the synthesis and analysis of regularized primal-dual gradient methods to track a Karush-Kuhn-Tucker (KKT) trajectory. The proposed regularized primal-dual gradient methods are implemented in a running fashion, in the sense that the underlying optimization problem changes during the iterations of the algorithms. For a problem with twice continuously differentiable cost and constraints, and under a generalization of the Mangasarian-Fromovitz constraint qualification, sufficient conditions are derived for the running algorithm to track a KKT trajectory. Further, asymptotic bounds for the tracking error (as a function of the time-variability of a KKT trajectory) are obtained. A continuous-time version of the algorithm, framed as a system of differential inclusions, is also considered and analytical convergence results are derived. For the continuous-time setting, a set of sufficient conditions for the KKT trajectories not to bifurcate or merge is proposed. Illustrative numerical results inspired by a real-world application are provided.

Mordukhovich stationarity for mathematical programs with switching constraints under weak constraint qualifications

The mathematical program with switching constraints (MPSC), which has been introduced recently, is a difficult class of optimization problems since standard constraint qualifications are very likely to fail at local minimizers. Due to the failure of standard constraint qualifications, it is reasonable to propose some constraint qualifications for local minimizers to satisfy some stationarity conditions that are generally weaker than Karush-Kuhn-Tucker stationarity such as Mordukhovich (M-) stationarity. First, we propose the weakest constraint qualification for M-stationarity of MPSC to hold at local minimizers. Then we extend some weak verifiable constraint qualifications for nonlinear programming to allow the existence of switching constraints, which are all strictly weaker than MPSC linear independence constraint qualification and/or MPSC Mangasarian-Fromovitz constraint qualification used in the literature. We show that most of the newly introduced constraint qualifications are sufficient for local minimizers to be M-stationary. Finally, the relations among MPSC tailored constraint qualifications are discussed.

Constraint qualifications in terms of convexificators for nonsmooth programming problems with mixed constraints

The main aim of the paper is to introduce certain constraint qualifications for a nonsmooth programming problem in terms of semi-regular convexificators and investigate their relations with other existing notions of constraint qualifications. The programming problem under consideration has mixed constraints, that is, it involves both inequality and equality constraints. All these notions are in terms of upper semi-regular convexificators of inequality constraints and pseudo-differentials of equality constraints. Based on a sufficient condition for error bound property, the implication relation between quasinormality and error bound property in terms of convexificators is investigated in this paper. Three conditions are introduced, namely constant positive linear dependence condition (CPLD), constant rank constraint qualification (CRCQ) and Mangasarian–Fromovitz constraint qualification (MFCQ) in terms of convexificators. These conditions are in fact shown to be constraint qualifications as Karush–Kuhn–Tucker optimality conditions hold when CPLD holds and both MFCQ and CRCQ imply CPLD. Further, it is observed that CPLD and quasinormality conditions are independent for nonsmooth problems in terms of convexificators.

On the strong metric subregularity in mathematical programming

Abstract This note presents sufficient conditions for the property of strong metric subregularity (SMSr) of the system of first order optimality conditions for a mathematical programming problem in a Banach space (the Karush-Kuhn-Tucker conditions). The constraints of the problem consist of equations in a Banach space setting and a finite number of inequalities. The conditions, under which SMSr is proven, assume that the data are twice continuously Fréchet differentiable, the strict Mangasarian-Fromovitz constraint qualification is satisfied, and the second-order sufficient optimality condition holds. The obtained result extends the one known for finite-dimensional problems. Although the applicability of the result is limited to the Banach space setting (due to the twice Fréchet differentiability assumptions and the finite number of inequality constraints), the paper can be valuable due to the self-contained exposition, and provides a ground for extensions. One possible extension was recently implemented in Osmolovskii and Veliov (2021).

Directional Necessary Optimality Conditions for Bilevel Programs

The bilevel program is an optimization problem in which the constraint involves solutions to a parametric optimization problem. It is well known that the value function reformulation provides an equivalent single-level optimization problem, but it results in a nonsmooth optimization problem that never satisfies the usual constraint qualification, such as the Mangasarian–Fromovitz constraint qualification (MFCQ). In this paper, we show that even the first order sufficient condition for metric subregularity (which is, in general, weaker than MFCQ) fails at each feasible point of the bilevel program. We introduce the concept of a directional calmness condition and show that, under the directional calmness condition, the directional necessary optimality condition holds. Although the directional optimality condition is, in general, sharper than the nondirectional one, the directional calmness condition is, in general, weaker than the classical calmness condition and, hence, is more likely to hold. We perform the directional sensitivity analysis of the value function and propose the directional quasi-normality as a sufficient condition for the directional calmness. An example is given to show that the directional quasi-normality condition may hold for the bilevel program.

Normality, Controllability and Properness in Optimal Control

In this paper we show that, for optimal control problems involving equality and inequality constraints on the control function, the notions of normality and properness (or the Mangasarian–Fromovitz constraint qualification) of a trajectory relative to the set of constraints are equivalent. This is in contrast with some differences recently obtained between the theories of mathematical programming and optimal control, and it provides an important insight in the derivation of first and second order necessary optimality conditions for infinite dimensional problems.

The Supporting Role of the Mangasarian-Fromovitz Constraint Qualification in Calculus of Variations

The Mangasarian-Fromovitz constraint qualification has played a fundamental role in mathematical programming problems involving inequality constraints. It is known to be equivalent to a normality condition (in terms of the positive linear independence of active gradients) which, in turn, implies regularity (the tangent and the linearizing cones coincide), a condition which has been crucial in the derivation of first- and second-order necessary optimality conditions. In this paper, we study the corresponding implications for problems in the calculus of variations. In particular, we show how the equivalence between normality and the Mangasarian-Fromovitz constraint qualification is preserved, but also that their main role changes completely since, as a simple example shows, they may not imply the corresponding regularity condition.

Necessary optimality conditions in nonsmooth semi-infinite multiobjective optimization under metric subregularity

We consider nonsmooth semi-infinite multiobjective optimization problems under mixed constraints, including infinitely many mixed constraints by using Clarke subdifferential. Semi-infinite programming (SIP) is the minimization of many scalar objective functions subject to a possibly infinite system of inequality or/and equality constraints. SIPs have been proved to be very important in optimization and applications. Semi-infinite programming problems arise in various fields of engineering such as control systems design, decision making under competition, and multiobjective optimization. There is extensive literature on standard semi-infinite programming problems. The investigation of optimality conditions for these problems is always one of the most attractive topics and has been studied extensively in the literature. In our work, we study optimality conditions for weak efficiency of a multiobjective semi-infinite optimization problem under mixed constraints including infinitely many of both equality and inequality constraints in terms of Clarke subdifferential. Our conditions are the form of the Karush-Kuhn-Tucker (KKT) multiplier. To the best of our knowledge, only a few papers are dealing with optimality conditions for SIPs subject to mixed constraints. By the Pshenichnyi-Levin-Valadire (PLV) property and the directional metric subregularity, we introduce a type of Mangasarian-Fromovitz constraint qualification (MFCQ). Then we show that (MFCQ) is a sufficient condition to guarantee the extended Abadie constraint qualification (ACQ) to satisfy. In our constraint qualifications, all functions are nonsmooth and the number of constraints is not necessarily finite. In our paper, we do not need the involved functions: convexity and differentiability. Later, we apply the extended Abadie constraint qualification to get the KKT multipliers for weak efficient solutions of SIP. Many examples are provided to illustrate some advantages of our results. The paper is organized as follows. In Section Preliminaries, we present our basic definitions of nonsmooth and convex analysis. Section Main Results prove necessary conditions for the weakly efficient solution in terms of the Karush-Kuhn-Tucker mult iplier rule with the help of some constraint qualifications.

A Comparison of Normal Cone Conditions for Homotopy Methods for Solving Inequality Constrained Nonlinear Programming Problems

Homotopy methods are powerful tools for solving nonlinear programming. Their global convergence can be generally established under conditions of the nonemptiness and boundness of the interior of the feasible set, the Positive Linear Independent Constraint Qualification (PLICQ), which is equivalent to the Mangasarian-Fromovitz Constraint Qualification (MFCQ), and the normal cone condition. This paper provides a comparison of the existing normal cone conditions used in homotopy methods for solving inequality constrained nonlinear programming.

A Sequential Quadratic Programming Method for Constrained Multi-objective Optimization Problems

In this article, a globally convergent sequential quadratic programming (SQP) method is developed for multi-objective optimization problems with inequality type constraints. A feasible descent direction is obtained using a linear approximation of all objective functions as well as constraint functions. The sub-problem at every iteration of the sequence has feasible solution. A non-differentiable penalty function is used to deal with constraint violations. A descent sequence is generated which converges to a critical point under the Mangasarian-Fromovitz constraint qualification along with some other mild assumptions. The method is compared with a selection of existing methods on a suitable set of test problems.

Relaxed constant positive linear dependence constraint qualification and its application to bilevel programs

Relaxed constant positive linear dependence constraint qualification (RCPLD) for a system of smooth equalities and inequalities is a constraint qualification that is weaker than the usual constraint qualifications such as Mangasarian Fromovitz constraint qualification and the linear constraint qualification. Moreover RCPLD is known to induce an error bound property. In this paper we extend RCPLD to a very general feasibility system which may include Lipschitz continuous inequality constraints, complementarity constraints and abstract constraints. We show that this RCPLD for the general system is a constraint qualification for the optimality condition in terms of limiting subdifferential and limiting normal cone and it is a sufficient condition for the error bound property under the strict complementarity condition for the complementarity system and Clarke regularity conditions for the inequality constraints and the abstract constraint set. Moreover we introduce and study some sufficient conditions for RCPLD including the relaxed constant rank constraint qualification. Finally we apply our results to the bilevel program.

Fast sensitivity-based economic model predictive control for degenerate systems

We present a sensitivity-based nonlinear model predictive control (NMPC) algorithm and demonstrate it on a case study with an economic cost function. In contrast to existing sensitivity-based approaches that make strong assumptions on the underlying optimization problem (e.g. the linear independence constraint qualification implying unique multiplier), our method is designed to handle problems satisfying a weaker constraint qualification, namely the Mangasarian-Fromovitz constraint qualification (MFCQ). Our nonlinear programming (NLP) sensitivity update consists of three steps. The first step is a corrector step in which a system of linear equations is solved. Then a predictor step is computed by a quadratic program (QP). Finally, a linear program (LP) is solved to select the multipliers that give the correct sensitivity information. A path-following scheme containing these steps is embedded in the advanced-step NMPC (asNMPC) framework. We demonstrate our method on a large-scale case example consisting of a reactor and distillation process. We show that LICQ does not hold and the path-following method is able to accurately approximate the ideal solutions generated by an NLP solver.

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.