New perspective on the Kuhn–Tucker theorem

A new proof of the Kuhn–Tucker theorem on necessary conditions for a minimum of a differentiable function of several variables in the case of inequality constraints is given. The proof relies on a simple inequality (common in textbooks) for the projection of a vector onto a convex set.     

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.     

A simple and elementary proof of the Karush–Kuhn–Tucker theorem for inequality-constrained optimization

We present an elementary proof of the Karush–Kuhn–Tucker Theorem for the problem with nonlinear inequality constraints and linear equality constraints. Most proofs in the literature rely on advanced optimization concepts such as linear programming duality, the convex separation theorem, or a theorem of the alternative for systems of linear inequalities. By contrast, the proof given here uses only basic facts from linear algebra and the definition of differentiability.     

Pareto Optimizing and Kuhn–Tucker Stationary Sequences

We study the relation between weakly Pareto minimizing and Kuhn–Tucker stationary nonfeasible sequences for vector optimization under constraints, where the weakly Pareto (efficient) set may be empty. The work is placed in a context of Banach spaces and the constraints are described by a functional taking values in a cone. We characterize the asymptotic feasibility in terms of the constraint map and the asymptotic efficiency via a Kuhn–Tucker system completely approximate, distinguishing the classical bounded case from the nontrivial unbounded one. The latter requires Auslender–Crouzeix type conditions and Ekeland's variational principle for constrained vector problems.     

A New Proof of the Assmus–Mattson Theorem for Non-Binary Codes

C. Bachoc gave a new proof of theAssmus–Mattson theorem for linear binary codes using harmonicweight enumerators which she defined B. We give a new proof ofthe Assmus–Mattson theorem for linear codes over any finitefield using similar methods.     

Enhanced Karush–Kuhn–Tucker condition and weaker constraint qualifications

In this paper we study necessary optimality conditions for nonsmooth optimization problems with equality, inequality and abstract set constraints. We derive the enhanced Fritz John condition which contains some new information even in the smooth case than the classical enhanced Fritz John condition. From this enhanced Fritz John condition we derive the enhanced Karush–Kuhn–Tucker condition and introduce the associated pseudonormality and quasinormality condition. We prove that either pseudonormality or quasinormality with regularity on the constraint functions and the set constraint implies the existence of a local error bound. Finally we give a tighter upper estimate for the Fréchet subdifferential and the limiting subdifferential of the value function in terms of quasinormal multipliers which is usually a smaller set than the set of classical normal multipliers. In particular we show that the value function of a perturbed problem is Lipschitz continuous under the perturbed quasinormality condition which is much weaker than the classical normality condition.     

A New Proof of Calabi-Yau's Theorem

We give a new proof of Calabi-Yau's theorem on the volume growth of Rie- mannian manifolds with non-negative Ricci curvature.     

Karush–Kuhn–Tucker Conditions in Set Optimization

This paper investigates set optimization problems in finite dimensional spaces with the property that the images of the set-valued objective map are described by inequalities and equalities and that sets are compared with the set less order relation. For these problems new Karush–Kuhn–Tucker conditions are shown as necessary and sufficient optimality conditions. Optimality conditions without multiplier of the objective map are also presented. The usefulness of these results is demonstrated with a standard example.     

A Generalization of the Karush–Kuhn–Tucker Theorem for Approximate Solutions of Mathematical Programming Problems Based on Quadratic Approximation

In computations related to mathematical programming problems, one often has to consider approximate, rather than exact, solutions satisfying the constraints of the problem and the optimality criterion with a certain error. For determining stopping rules for iterative procedures, in the stability analysis of solutions with respect to errors in the initial data, etc., a justified characteristic of such solutions that is independent of the numerical method used to obtain them is needed. A necessary δ-optimality condition in the smooth mathematical programming problem that generalizes the Karush–Kuhn–Tucker theorem for the case of approximate solutions is obtained. The Lagrange multipliers corresponding to the approximate solution are determined by solving an approximating quadratic programming problem.     

A Proof of the Beyer and Stein Comjecture

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A New Proof for the Interpolating Theorem

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A New Proof of Diophantine Equation ■

By using algebraic number theory and p-adic analysis method,we give a new and simple proof of Diophantine equation ■.     

Nonsmooth multiobjective programming: strong Kuhn–Tucker conditions

We consider a multiobjective optimization problem with a feasible set defined by inequality and equality constraints and a set constraint, where the objective and constraint functions are locally Lipschitz. Several constraint qualifications are given in such a way that they generalize the classical ones, when the functions are differentiable. The relationships between them are analyzed. Then, we establish strong Kuhn–Tucker necessary optimality conditions in terms of the Clarke subdifferentials such that the multipliers of the objective function are all positive. Furthermore, sufficient optimality conditions under generalized convexity assumptions are derived. Moreover, the concept of efficiency is used to formulate duality for nonsmooth multiobjective problems. Wolf and Mond–Weir type dual problems are formulated. We also establish the weak and strong duality theorems.     

Solving the Karush–Kuhn–Tucker system of a nonconvex programming problem on an unbounded set

In the papers [G.C. Feng, B. Yu, Combined homotopy interior point method for nonlinear programming problems, in: H. Fujita, M. Yamaguti (Eds.), Advances in Numerical Mathematics; Proceedings of the Second Japan–China Seminar on Numerical Mathematics, in: Lecture Notes in Numerical and Applied Analysis, vol. 14, Kinokuniya, Tokyo, 1995, pp. 9–16; G.C. Feng, Z.H. Lin, B. Yu, Existence of an interior pathway to a Karush–Kuhn–Tucker point of a nonconvex programming problem, Nonlinear Analysis 32 (1998) 761–768; Z.H. Lin, B. Yu, G.C. Feng, A combined homotopy interior point method for convex programming problem, Applied Mathematics and Computation 84 (1997) 193–211], a combined homotopy interior method was presented and global convergence results obtained for nonconvex nonlinear programming when the feasible set is bounded and satisfies the so called normal cone condition. However, for when the feasible set is not bounded, no result has so far been obtained. In this paper, a combined homotopy interior method for nonconvex programming problems on the unbounded feasible set is considered. Under suitable additional assumptions, boundedness of the homotopy path, and hence global convergence, is proven.     

Properties of equation reformulation of the Karush–Kuhn–Tucker condition for nonlinear second order cone optimization problems

We give an equation reformulation of the Karush–Kuhn–Tucker (KKT) condition for the second order cone optimization problem. The equation is strongly semismooth and its Clarke subdifferential at the KKT point is proved to be nonsingular under the constraint nondegeneracy condition and a strong second order sufficient optimality condition. This property is used in an implicit function theorem of semismooth functions to analyze the convergence properties of a local sequential quadratic programming type (for short, SQP-type) method by Kato and Fukushima (Optim Lett 1:129–144, 2007). Moreover, we prove that, a local solution x* to the second order cone optimization problem is a strict minimizer of the Han penalty merit function when the constraint nondegeneracy condition and the strong second order optimality condition are satisfied at x*.     

On Weak and Strong Kuhn–Tucker Conditions for Smooth Multiobjective Optimization

We consider a smooth multiobjective optimization problem with inequality constraints. Weak Kuhn?CTucker (WKT) optimality conditions are said to hold for such problems when not all the multipliers of the objective functions are zero, while strong Kuhn?CTucker (SKT) conditions are said to hold when all the multipliers of the objective functions are positive. We introduce a new regularity condition under which (WKT) hold. Moreover, we prove that for another new regularity condition (SKT) hold at every Geoffrion-properly efficient point. We show with an example that the assumption on proper efficiency cannot be relaxed. Finally, we prove that Geoffrion-proper efficiency is not needed when the constraint set is polyhedral and the objective functions are linear.     

On Classes of Generalized Convex Functions, Gordan–Farkas Type Theorems, and Lagrangian Duality

In this paper, we introduce several classes of generalized convex functions already discussed in the literature and show the relation between these classes. Moreover, a Gordan–Farkas type theorem is proved for all these classes and it is shown how these theorems can be used to verify strong Lagrangian duality results in finite-dimensional optimization.     

A New Proof of Calabi-Yau＇s Theorem

We give a new proof of Calabi-Yau＇s theorem on the volume growth of Riemannian manifolds with non-negative Ricci curvature.