Existence and Boundedness of the Kuhn-Tucker Multipliers in Nonsmooth Multiobjective Optimization

Using the idea of upper convexificators, we propose constraint qualifications and study existence and boundedness of the Kuhn-Tucker multipliers for a nonsmooth multiobjective optimization problem with inequality constraints and an arbitrary set constraint. We show that, at locally weak efficient solutions where the objective and constraint functions are locally Lipschitz, the constraint qualifications are necessary and sufficient conditions for the Kuhn-Tucker multiplier sets to be nonempty and bounded under certain semiregularity assumptions on the upper convexificators of the functions.     

Constraint qualifications and Lagrange multipliers in nondifferentiable programming problems

In this paper, we present several constraint qualifications, and we show that these conditions guarantee the nonvacuity and the boundedness of the Lagrange multiplier sets for general nondifferentiable programming problems. The relationships with various constraint qualifications are investigated.The author gratefully acknowledges the comments made by the two referees.     

On Regularity for Constrained Extremum Problems. Part 2: Necessary Optimality Conditions

A particular theorem for linear separation between two sets is applied in the image space associated with a constrained extremum problem. In this space, the two sets are a convex cone, depending on the constraints (equalities and inequalities) of the given problem and the homogenization of its image. It is proved that the particular linear separation is equivalent to the existence of Lagrangian multipliers with a positive multiplier associated with the objective function (i.e., a necessary optimality condition). A comparison with the constraint qualifications and the regularity conditions existing in the literature is performed.     

Unified theory of augmented Lagrangian methods for constrained global optimization

We classify in this paper different augmented Lagrangian functions into three unified classes. Based on two unified formulations, we construct, respectively, two convergent augmented Lagrangian methods that do not require the global solvability of the Lagrangian relaxation and whose global convergence properties do not require the boundedness of the multiplier sequence and any constraint qualification. In particular, when the sequence of iteration points does not converge, we give a sufficient and necessary condition for the convergence of the objective value of the iteration points. We further derive two multiplier algorithms which require the same convergence condition and possess the same properties as the proposed convergent augmented Lagrangian methods. The existence of a global saddle point is crucial to guarantee the success of a dual search. We generalize in the second half of this paper the existence theorems for a global saddle point in the literature under the framework of the unified classes of augmented Lagrangian functions.     

具强制条件的向量均衡问题

研究锥伪单调、锥拟凸和上锥连续映射在某种强制性条件下的向量均衡问题解集的特征,建立强制性条件与向量均衡问题解集的关系,得到对偶向量均衡问题局部解集含于向量均衡问题解集的性质和向量均衡问题解集的非空性条件,给出在锥伪单调、锥拟凸和上锥连续映射条件下向量均衡问题解集的非空有界性与强制性条件的等价性．     

Constraint Qualifications Characterizing Lagrangian Duality in Convex Optimization

In convex optimization, a constraint qualification (CQ) is an essential ingredient for the elegant and powerful duality theory. Various constraint qualifications which are sufficient for the Lagrangian duality have been given in the literature. In this paper, we present constraint qualifications which characterize completely the Lagrangian duality.     

Boundedness and Nonemptiness of the Efficient Solution Sets in Multiobjective Optimization

For a given multiobjective optimization problem, we study recession properties of the sets of efficient solutions and properly efficient solutions. We work out various consequences based on the obtained recession properties, including a characterization for the boundedness and nonemptiness of the set of (properly) efficient solutions when the problem is a convex problem. We also show that the boundedness and nonemptiness of the set of efficient solutions is equivalent to that of the set of properly efficient solutions under an additional mild condition. Finally, we provide some new verifiable necessary conditions for the nonemptiness of the set of efficient solutions in terms of the associated recession functions and recession cones.     

Global Convergence of Algorithms Under Constant Rank Conditions for Nonlinear Second-Order Cone Programming

In Andreani et al. (Weak notions of nondegeneracy in nonlinear semidefinite programming, 2020), the classical notion of nondegeneracy (or transversality) and Robinson’s constraint qualification have been revisited in the context of nonlinear semidefinite programming exploiting the structure of the problem, namely its eigendecomposition. This allows formulating the conditions equivalently in terms of (positive) linear independence of significantly smaller sets of vectors. In this paper, we extend these ideas to the context of nonlinear second-order cone programming. For instance, for an m-dimensional second-order cone, instead of stating nondegeneracy at the vertex as the linear independence of m derivative vectors, we do it in terms of several statements of linear independence of 2 derivative vectors. This allows embedding the structure of the second-order cone into the formulation of nondegeneracy and, by extension, Robinson’s constraint qualification as well. This point of view is shown to be crucial in defining significantly weaker constraint qualifications such as the constant rank constraint qualification and the constant positive linear dependence condition. Also, these conditions are shown to be sufficient for guaranteeing global convergence of several algorithms, while still implying metric subregularity and without requiring boundedness of the set of Lagrange multipliers.

     

A note on upper Lipschitz stability, error bounds, and critical multipliers for Lipschitz-continuous KKT systems

We prove a new local upper Lipschitz stability result and the associated local error bound for solutions of parametric Karush–Kuhn–Tucker systems corresponding to variational problems with Lipschitzian base mappings and constraints possessing Lipschitzian derivatives, and without any constraint qualifications. This property is equivalent to the appropriately extended to this nonsmooth setting notion of noncriticality of the Lagrange multiplier associated to the primal solution, which is weaker than second-order sufficiency. All this extends several results previously known only for optimization problems with twice differentiable data, or assuming some constraint qualifications. In addition, our results are obtained in the more general variational setting.     

The Tangent cones on constraint qualifications in optimization problems

This article proposes a few tangent cones,which are relative to the constraint qualifications of optimization problems.With the upper and lower directional derivatives of an objective function,the characteristics of cones on the constraint qualifications are presented.The interrelations among the constraint qualifications,a few cones involved, and level sets of upper and lower directional derivatives are derived.     

An Exact Augmented Lagrangian Function for Nonlinear Programming with Two-Sided Constraints

This paper is aimed toward the definition of a new exact augmented Lagrangian function for two-sided inequality constrained problems. The distinguishing feature of this augmented Lagrangian function is that it employs only one multiplier for each two-sided constraint. We prove that stationary points, local minimizers and global minimizers of the exact augmented Lagrangian function correspond exactly to KKT pairs, local solutions and global solutions of the constrained problem.     

Existence results for a class of hemivariational inequality problems on Hadamard manifolds

In this paper, a class of hemivariational inequality problems are introduced and studied on Hadamard manifolds. Using the properties of Clarke’s generalized directional derivative and Fan-KKM lemma, an existence theorem of solution in connection with the hemivariational inequality problem is obtained when the constraint set is bounded. By employing some coercivity conditions and the properties of Clarke’s generalized directional derivative, an existence result and the boundedness of the set of solutions for the underlying problem are investigated when the constraint set is unbounded. Moreover, a sufficient and necessary condition for ensuring the nonemptiness of the set of solutions concerned with the hemivariational inequality problem is also given.     

Regularized Lagrangian duality for linearly constrained quadratic optimization and trust-region problems

In this paper we first establish a Lagrange multiplier condition characterizing a regularized Lagrangian duality for quadratic minimization problems with finitely many linear equality and quadratic inequality constraints, where the linear constraints are not relaxed in the regularized Lagrangian dual. In particular, in the case of a quadratic optimization problem with a single quadratic inequality constraint such as the linearly constrained trust-region problems, we show that the Slater constraint qualification (SCQ) is necessary and sufficient for the regularized Lagrangian duality in the sense that the regularized duality holds for each quadratic objective function over the constraints if and only if (SCQ) holds. A new theorem of the alternative for systems involving both equality constraints and two quadratic inequality constraints plays a key role. We also provide classes of quadratic programs, including a class of CDT-subproblems with linear equality constraints, where (SCQ) ensures regularized Lagrangian duality.     

Optimality conditions for nondifferentiable convex semi-infinite programming

This paper gives characterizations of optimal solutions to the nondifferentiable convex semi-infinite programming problem, which involve the notion of Lagrangian saddlepoint. With the aim of giving the necessary conditions for optimality, local and global constraint qualifications are established. These constraint qualifications are based on the property of Farkas-Minkowski, which plays an important role in relation to certain systems obtained by linearizing the feasible set. It is proved that Slater's qualification implies those qualifications.     

pth Power Lagrangian Method for Integer Programming

When does there exist an optimal generating Lagrangian multiplier vector (that generates an optimal solution of an integer programming problem in a Lagrangian relaxation formulation), and in cases of nonexistence, can we produce the existence in some other equivalent representation space? Under what conditions does there exist an optimal primal-dual pair in integer programming? This paper considers both questions. A theoretical characterization of the perturbation function in integer programming yields a new insight on the existence of an optimal generating Lagrangian multiplier vector, the existence of an optimal primal-dual pair, and the duality gap. The proposed pth power Lagrangian method convexifies the perturbation function and guarantees the existence of an optimal generating Lagrangian multiplier vector. A condition for the existence of an optimal primal-dual pair is given for the Lagrangian relaxation method to be successful in identifying an optimal solution of the primal problem via the maximization of the Lagrangian dual. The existence of an optimal primal-dual pair is assured for cases with a single Lagrangian constraint, while adopting the pth power Lagrangian method. This paper then shows that an integer programming problem with multiple constraints can be always converted into an equivalent form with a single surrogate constraint. Therefore, success of a dual search is guaranteed for a general class of finite integer programming problems with a prominent feature of a one-dimensional dual search.     

Existence and Boundedness of Solutions in Infinite-Dimensional Vector Optimization Problems

This work focuses on the nonemptiness and boundedness of the sets of efficient and weak efficient solutions of a vector optimization problem, where the decision space is a normed space and the image space is a locally convex Hausdorff topological linear space. By studying certain boundedness and coercivity concepts of vector-valued functions and via an asymptotic analysis, we extend to this kind of problems some well-known existence and boundedness results for efficient and weak efficient solutions of multiobjective optimization problems with Pareto or polyhedral orderings. Some of these results are proved under weaker assumptions.     

A dual algorithm for the one-machine scheduling problem

A branch and bound algorithm is presented for the problem of schedulingn jobs on a single machine to minimize tardiness. The algorithm uses a dual problem to obtain a good feasible solution and an extremely sharp lower bound on the optimal objective value. To derive the dual problem we regard the single machine as imposing a constraint for each time period. A dual variable is associated with each of these constraints and used to form a Lagrangian problem in which the dualized constraints appear in the objective function. A lower bound is obtained by solving the Lagrangian problem with fixed multiplier values. The major theoretical result of the paper is an algorithm which solves the Lagrangian problem in a number of steps proportional to the product ofn ² and the average job processing time. The search for multiplier values which maximize the lower bound leads to the formulation and optimization of the dual problem. The bounds obtained are so sharp that very little enumeration or computer time is required to solve even large problems. Computational experience with 20-, 30-, and 50-job problems is presented.     

Strict Feasibility for Generalized Mixed Variational Inequality in Reflexive Banach Spaces

The purpose of this paper is to investigate the nonemptiness and boundedness of solution set for a generalized mixed variational inequality with strict feasibility in reflexive Banach spaces. A concept of strict feasibility for the generalized mixed variational inequality is introduced, which recovers the existing concepts of strict feasibility for variational inequalities and complementarity problems. By using the equivalence characterization of nonemptiness and boundedness of the solution set for the generalized mixed variational inequality due to Zhong and Huang (J. Optim. Theory Appl. 147:454–472, 2010), it is proved that the generalized mixed variational inequality problem has a nonempty bounded solution set is equivalent to its strict feasibility.     

An extension of the Basic Constraint Qualification to nonconvex vector optimization problems

In this paper a Basic Constraint Qualification is introduced for a nonconvex infinite-dimensional vector optimization problem extending the usual one from convex programming assuming the Hadamard differentiability of the maps. Corresponding KKT conditions are established by considering a decoupling of the constraint cone into half-spaces. This extension leads to generalized KKT conditions which are finer than the usual abstract multiplier rule. A second constraint qualification expressed directly in terms of the data is also introduced, which allows us to compute the contingent cone to the feasible set and, as a consequence, it is proven that this condition is a particular case of the first one. Relationship with other constraint qualifications in infinite-dimensional vector optimization, specially with the Kurcyuscz-Robinson-Zowe constraint qualification, are also given.