Constraint Qualification Condition Research Articles

The Toland-Fenchel-Lagrange duality of DC programs for composite convex functions

In this paper, by virtue of the epigraph technique, we construct a new kind of closedness-type constraintqualification, which is the sufficient and necessary condition to guarantee thestrong duality between a cone constraint composite optimization problem and its dual problem holds. Under this closedness-type constraint qualification condition, we obtain a formula of subdifferential for composite functions and study a cone constraint composite DC optimization problem in locally convex Hausdorff topological vector spaces.

On Metric Pseudo-(sub)Regularity of Multifunctions and Optimality Conditions for Degenerated Mathematical Programs

This paper is devoted to the analysis of a special kind of regularity of a multifunction which we call metric pseudo-(sub)regularity, when the multifunction is not metrically (sub)regular at a given point but is metrically (sub)regular at certain points in a neighborhood with moduli possibly tending to infinity with a certain order. By using advanced techniques of generalized differentiation we derive conditions both necessary and sufficient for this property. As a byproduct we obtain a new sufficient condition for metric subregularity. Then we apply these results to multifunctions composed by a smooth mapping and a generalized polyhedral multifunction and obtain explicit formulas for this case. Finally we show how the theory can be used to obtain necessary optimality conditions when the constraint qualification condition of metric (sub)regularity is violated.

Optimality conditions and finite convergence of Lasserre’s hierarchy

Lasserre’s hierarchy is a sequence of semidefinite relaxations for solving polynomial optimization problems globally. This paper studies the relationship between optimality conditions in nonlinear programming theory and finite convergence of Lasserre’s hierarchy. Our main results are: (i) Lasserre’s hierarchy has finite convergence when the constraint qualification, strict complementarity and second order sufficiency conditions hold at every global minimizer, under the standard archimedean condition; the proof uses a result of Marshall on boundary hessian conditions. (ii) These optimality conditions are all satisfied at every local minimizer if a finite set of polynomials, which are in the coefficients of input polynomials, do not vanish at the input data (i.e., they hold in a Zariski open set). This implies that, under archimedeanness, Lasserre’s hierarchy has finite convergence generically.

Convergence analysis of a regularized interior point algorithm for the barrier problems with singular solutions

Wächter and Biegler (2006) [25] presented and implemented an interior point filter line search method (called IPOPT) for large scale nonlinear programming. To make IPOPT efficient and robust, they add some safeguards in their implementation. One crucial safeguard is to apply the inertia correction technique to modify the KKT matrix. We focus on the degenerate problems in which the KKT matrix might be singular. To complement theoretical foundation of this technique in the degenerate case, we propose and analyze a regularized interior point method for the barrier problems with singular solutions which largely relies on IPOPT. Based on the inertia correction technique, the matrix arising in the KKT equation is corrected to be a nonsingular one so that the Newton step can be calculated at each iteration even in the degenerate case. We emphasize that the special inertia correction for the KKT matrix does not impose any negative impact on the fast convergence of our algorithm. Under some local error bound condition, we prove that the rate of quadratic convergence is achieved when the second order condition or some constraint qualification condition fails to hold (or both).

Infinite Alternative Theorems and Nonsmooth Constraint Qualification Conditions

In this paper, some infinite alternative theorems in locally convex vector spaces are proved and some applications of these theorems in optimization theory are addressed. Extensions of constraint qualification conditions, in the absence of differentiability using Clarke’s generalized gradient and Mordukhovich’s subdifferential, in Banach and Asplund spaces are introduced; and relationships between them are established. Some of these relations, are proved using the provided alternative theorems.

On the Convergence of a Smooth Penalty Algorithm without Computing Global Solutions

We consider a smooth penalty algorithm to solve nonconvex optimization problem based on a family of smooth functions that approximate the usual exact penalty function. At each iteration in the algorithm we only need to find a stationary point of the smooth penalty function, so the difficulty of computing the global solution can be avoided. Under a generalized Mangasarian‐Fromovitz constraint qualification condition (GMFCQ) that is weaker and more comprehensive than the traditional MFCQ, we prove that the sequence generated by this algorithm will enter the feasible solution set of the primal problem after finite times of iteration, and if the sequence of iteration points has an accumulation point, then it must be a Karush‐Kuhn‐Tucker (KKT) point. Furthermore, we obtain better convergence for convex optimization problem.

Subdifferential of Optimal Value Functions in Nonlinear Infinite Programming

This paper presents an exact formula for computing the normal cones of the constraint set mapping including the Clarke normal cone and the Mordukhovich normal cone in infinite programming under the extended Mangasarian-Fromovitz constraint qualification condition. Then, we derive an upper estimate as well as an exact formula for the limiting subdifferential of the marginal/optimal value function in a general Banach space setting.

ON THE GLOBAL CONVERGENCE OF A MODIFIED SEQUENTIAL QUADRATIC PROGRAMMING ALGORITHM FOR NONLINEAR PROGRAMMING PROBLEMS WITH INEQUALITY CONSTRAINTS

When a Sequential Quadratic Programming (SQP) method is used to solve the nonlinear programming problems, one of the main difficulties is that the Quadratic Programming (QP) subproblem may be incompatible. In this paper, an SQP algorithm is given by modifying the traditional QP subproblem and applying a class of penalty function whose penalty parameters can be adjusted automatically. The new QP subproblem is compatible. Under the extended Mangasarian-Fromovitz constraint qualification condition and the boundedness of the iterates, the algorithm is showed to be globally convergent to a KKT point of the non-linear programming problem.

Efficiency for multiobjective multidisciplinary optimization problems with quasiseparable subproblems

Multidisciplinary optimization (MDO) has proved to be a useful tool for engineering design problems. Multiobjective optimization has been introduced to strengthen MDO techniques and deal with non-comparable and conflicting design objectives. A large majority of papers on multiobjective MDO have been applied in nature. This paper develops theory of multiobjective MDO and examines relationships between efficient solutions of a quasi-separable multiobjective multidisciplinary optimization problem and efficient solutions of its separable counterpart. Equivalence of the original and separable problems in the context of the Kuhn-Tucker constraint qualification and efficiency conditions are proved. Two decomposition approaches are proposed and offer a possibility of finding efficient solutions of the original problem by only finding efficient solutions of the subproblems. The presented results are related to algorithms published in the engineering literature on multiobjective MDO.

Convex Inequalities Without Constraint Qualification nor Closedness Condition, and Their Applications in Optimization

Given two convex lower semicontinuous extended real valued functions F and h defined on locally convex spaces, we provide a dual transcription of the relation $$ F\left( 0,\cdot \right) \geq h\left( \cdot \right) . $$ (⋆) Some results in this direction are obtained in the first part of the paper (Lemma 2, Theorem 1). These results then are applied to the case when the left-hand-side in (⋆) is the sum of two convex functions with a convex composite one (Theorem 2). In the spirit of previous works (Hiriart-Urruty and Phelps, J Funct Anal 118:154–166, 1993; Penot, J Convex Anal 3:207–219, 1996, 2005; Thibault, 1995, SIAM J Control Optim 35:1434–1444, 1997, etc.) we give in Theorem 3 a formula for the subdifferential of such a function without any qualification condition. As a consequence of that, we extend to the nonreflexive setting a recent result (Jeyakumar et al., J Glob Optim 36:127–137 2006, Theorem 3.2) about subgradient optimality conditions without constraint qualifications. Finally, we apply Theorem 2 to obtain Farkas-type lemmas and new results on DC, convex, semi-definite, and linear optimization problems.

The Generalized Mangasarian-Fromowitz Constraint Qualification and Optimality Conditions for Bilevel Programs

We consider the optimal value reformulation of the bilevel programming problem. It is shown that the Mangasarian-Fromowitz constraint qualification in terms of the basic generalized differentiation constructions of Mordukhovich, which is weaker than the one in terms of Clarke’s nonsmooth tools, fails without any restrictive assumption. Some weakened forms of this constraint qualification are then suggested, in order to derive Karush-Kuhn-Tucker type optimality conditions for the aforementioned problem. Considering the partial calmness, a new characterization is suggested and the link with the previous constraint qualifications is analyzed.

Motion Control of a Container With Slosh: Constrained Sliding Mode Approach

A constrained sliding mode control methodology developed by the author is applied to the motion control of an open container filled with liquid. The objective is to control the position of the container to meet the performance and robustness requirements, and to specify a safe operating set, i.e., the set of initial conditions from which the system can be operated without exceeding a prescribed liquid level and additional constraints. A conventional sliding mode regulator is designed first to address nominal performance in the sliding mode. Then, a robustly invariant cylinder formed as the Cartesian product of an ellipsoid, and a closed interval is constructed using linear matrix inequalities. A set of constraint qualification conditions are evaluated to ensure that the intersection of the cylinder and the state constraints is robustly positively invariant, constituting the required operating set. Simulations and experimental results corresponding to a high-speed transfer validate the methodology.

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.

Representations of Positive Polynomials and Optimization on Noncompact Semialgebraic Sets

This paper studies the representation of a positive polynomial $f$ on a closed semialgebraic set $S:=\{x\in\mathbb{R}^n\mid g_i(x)=0, i=1,\dots,l, h_j(x)\geq0, j=1,\dots,m\}$ modulo the so-called critical ideal $I(f,S)$ of $f$ on $S$. Under a constraint qualification condition, it is demonstrated that, if either $f>0$ on $S$ or $f\geq0$ on $S$ and the critical ideal $I(f,S)$ is radical, then $f$ belongs to the preordering generated by the polynomials $h_1,\dots,h_m$ modulo the critical ideal $I(f,S)$. These facts imply that we can find a natural sequence of semidefinite programs whose optimal values converge monotonically, increasing to the infimum value $f^*:=\inf_{x\in S}f(x)$ of $f$ on $S$, provided that the infimum value is attained at some point. Besides, we shall construct a finite set in $\mathbb{R}$ containing the infimum value $f^*$. Moreover, some relations between the Fedoryuk [Soviet Math. Dokl., 17 (1976), pp. 486-490] and Malgrange [Complex Analysis, Microlocal Calculus and Relativistic Quantum Theory, Lecture Notes in Phys. 126, Springer, Berlin, 1980, pp. 170-177] conditions and coercivity for polynomials, which are bounded from below on $S$, are also established. In particular, a sufficient condition for $f$ to attain its infimum on $S$ is derived from these facts. We also show that every polynomial $f$, which is bounded from below on $S$, can be approximated in the $l_1$-norm of coefficients by a sequence of polynomials $f_\epsilon$ that are coercive. Finally, it is shown that almost every linear polynomial function, which is bounded from below on $S$, attains its infimum on $S$ and has the same asymptotic growth at infinity.

Solving polynomial optimization problems via the truncated tangency variety and sums of squares

Let f , g i , i = 1 , … , l , h j , j = 1 , … , m , be polynomials on R n and S ≔ { x ∈ R n ∣ g i ( x ) = 0 , i = 1 , … , l , h j ( x ) ≥ 0 , j = 1 , … , m } . This paper proposes a method for finding the global infimum of the polynomial f on the semialgebraic set S via sum of squares relaxation over its truncated tangency variety, even in the case where the polynomial f does not attain its infimum on S . Under a constraint qualification condition, it is demonstrated that: (i) The infimum of f on S and on its truncated tangency variety coincide; and (ii) A sums of squares certificate for nonnegativity of f on its truncated tangency variety. These facts imply that we can find a natural sequence of semidefinite programs whose optimal values converge, monotonically increasing to the infimum of f on S .

Effective method for constrained minimum — reverse bridge theorem

We propose reverse bridge theorem (RBTH) and give the demonstration in this work. RBTH is an effective method for constrained minimum. It is necessary and sufficient condition. RBTH bases on Extended saddle-point condition (ESPC), which is proposed by Yixin Chen. SGPlan according to ESPC dose the best in satisfying the International Planning Competition 2006 and in Suboptimal Metric Track of International Planning Competition 2004. RBTH comes into all advantage of ESPC and is different from ESPC, which can be concluded from two points. First, the core inequality of ESPC is formed by two sub-inequalities, and the core inequality of RBTH contains only one sub-inequality and one sub-equality. Generally, equality is easier to handle than inequality. RBTH should be better for planner than ESPC. The core inequality changes and is no longer saddle shape. We analyze why the core inequality changes. Second, continuous RBTH (CRBTH) does not need constraint-qualification condition which is necessary for continuous ESPC. This can be authenticated in two respects. One is the proof of CRBTH; the other is KKT which has relationship with ESPC. RBTH is real necessary and sufficient condition and ESPC is not. So RBTH can be used more widely. Besides, we show some errors in ESPC. We define point domain and Shrink-point domain to explain error. We also analyze the cause of errors, and provide an example.

Bounded sets of Lagrange multipliers for vector optimization problems in infinite dimension

The aim of this paper is to point out some sufficient constraint qualification conditions ensuring the boundedness of a set of Lagrange multipliers for vectorial optimization problems in infinite dimension. In some (smooth) cases these conditions turn out to be necessary for the existence of multipliers as well.

Duality and Lagrange multipliers for nonsmooth multiobjective programming

Without any constraint qualification, the necessary and sufficient optimality conditions are established in this paper for nonsmooth multiobjective programming involving generalised convex functions. With these optimality conditions, a mixed dual model is constructed which unifies two dual models. Several theorems on mixed duality and Lagrange multipliers are established in this paper.

AN EIGENVALUE PROBLEM FOR A CLASS OF NONLINEAR ELLIPTIC OPERATORS

Let a: [0, +∞) → [0, +∞) be an increasing continuous function with a(t) = 0 if and only if t = 0 and limt→+∞a(t) = +∞, Ω ⊂ ℝN be a bounded domain having the segment property and T[u,u] a nonnegative quadratic form involving the only generalized derivatives of order m of the function u: Ω → ℝ. Let p ≥ 1, μi ≠ 0 be real numbers, [Formula: see text], 1 ≤ i ≤ p and [Formula: see text] Put [Formula: see text] and [Formula: see text] Under certain hypotheses on Gi, we show that the minimization problem [Formula: see text] has a solution. Moreover, due to the well-known theorem on generalized multipliers involving the Robinson constraint qualification condition, the solution of the preceeding minimization problem is a weak solution of the corresponding Euler–Lagrange equation (1.1)–(1.2) below. We emphasize that no Δ2-condition on A or [Formula: see text] is imposed. One application to mechanics is given.

Surrogate Programming and Multipliers in Quasi-convex Programming

A result due to Luenberger on the existence of multipliers in a quasi-convex programming problem is extended to the case of constraints given by an arbitrary convex cone under a constraint qualification condition more general than Slater's condition. The existence of solutions is not assumed. We point out links with even convexity in the sense of Fenchel and quasi subdifferentiability in the sense of Greenberg--Pierskalla, and we observe that the couples of primal-dual optimal solutions reduce to saddle-points of a suitable Lagrangian function.