Lagrange Multiplier Rule Research Articles

Lagrange Multiplier Rules for Weakly Minimal Solutions of Compact-Valued Set Optimization Problems

Optimality conditions are established in terms of Lagrange–Fritz–John multipliers as well as Lagrange–Kuhn–Tucker multipliers for set optimization problems (without any convexity assumption) by using new scalarization techniques. Additionally, we indicate how these results may be applied to some particular weak vector equilibrium problems.

Lagrange Multiplier Rule to a Nonlinear Eigenvalue Problem in Musielak–Orlicz Spaces

In this paper, we shall be concerned with the existence result of the following eigenvalue problemin an arbitrary Musielak–Orlicz Spaces, with an indefinite bounded weight ρ which changes sign, and such that We show that the existence of an eigenvalue can be derived from a generalized version of Lagrange multiplier rule without assuming any restriction on generalized N-function M.

Characterizing the Lagrange multiplier rule in nonconvex set-valued optimization

In this article, by using the notions of contingent derivative, contingent epiderivative and generalized contingent epiderivative, we obtain some characterizations of the Lagrange multiplier rule at points which are not necessarily local minima.

Properties of the nonlinear scalar functional and its applications to vector optimization problems

The nonconvex separation theorem for the Hiriart-Urruty nonlinear scalar functional (Hiriart-Urruty in Math Oper Res 4:79–97, 1979) and its sublinearity and monotonicity with respect to convex cone have played an important role in the research of vector optimization problems. By means of these special properties, this functional also can be used as a coherent measure of an investment. This work is devoted to investigating the properties of this special scalarizing function with respect to a co-radiant set, which is more general than a cone and is also a main tool in the study of approximate solutions in vector optimization problems. We first show that although this scalar functional with respect to the co-radiant set is not necessarily positively homogeneous, it satisfies some special properties, which imply it is a co-radiant function and also has subadditivity and monotonicity under the convexity assumption. Based on the subadditivity property, we calculate the classical (Fenchel) subdifferentials for the scalar functional. Finally, as the applications of our results, we give the nonlinear scalar characterizations for the approximate solutions of vector optimization problems, and establish the Lagrange multiplier rules in term of the Mordukhovich subdifferential in Asplund spaces.

Affine Variational Inequalities on Normed Spaces

This paper studies infinite-dimensional affine variational inequalities on normed spaces. It is shown that infinite-dimensional quadratic programming problems and infinite-dimensional linear fractional vector optimization problems can be studied by using affine variational inequalities. We present two basic facts about infinite-dimensional affine variational inequalities: the Lagrange multiplier rule and the solution set decomposition.

Правило множителей Лагранжа для общей задачи на экстремум с бесконечным числом ограничений

Правило множителей Лагранжа для общей задачи на экстремум с бесконечным числом ограничений

Approximate solutions of vector optimization problems via improvement sets in real linear spaces

We deal with a constrained vector optimization problem between real linear spaces without assuming any topology and by considering an ordering defined through an improvement set E. We study E-optimal and weak E-optimal solutions and also proper E-optimal solutions in the senses of Benson and Henig. We relate these types of solutions and we characterize them through approximate solutions of scalar optimization problems via linear scalarizations and nearly E-subconvexlikeness assumptions. Moreover, in the particular case when the feasible set is defined by a cone-constraint, we obtain characterizations by means of Lagrange multiplier rules. The use of improvement sets allows us to unify and to extend several notions and results of the literature. Illustrative examples are also given.

Nonseparation of Sets and Optimality Conditions

Two new concepts of uniform approximating cones are introduced and discussed. The main result is a theorem for nonseparability of two closed sets. As an application of this result, an abstract Lagrange multiplier rule and a necessary optimality condition of Pontryagin maximum principle type for an optimal control problem in infinite-dimensional state space are obtained. These results allow one to treat infinite-dimensional optimal control problems with nonconvex target. The proposed approach reveals the importance of the uniformity of a tangent cone for obtaining necessary optimality conditions in an infinite-dimensional setting.

Inverse Functions and Existence Principles

Each one of six general existence principles of compactness (the extreme value theorem), completeness (the Newton method or the modified Newton method), topology (Brouwer’s fixed point theorem), homotopy (on contractions of a sphere to its center), variational analysis (Ekeland’s principle), and monotonicity (the Minty–Browder theorem)) is shown to lead to the inverse function theorem, each one giving some novel insight. There are differences in assumptions and algorithmic properties; some of the propositions have been constructed specially for this paper. Simple proofs of the last two principles are included. The proof by compactness is shorter and simpler than the shortest and simplest known proof, that by completion. This gives a very short self-contained proof of the Lagrange multiplier rule, which depends only on optimization methods. The proofs are of independent interest and are intended as well to be useful in the context of the ongoing efforts to obtain new variants of methods that are based on the inverse function theorem, such as comparative statics methods.

On the Proof of Pontryagin’s Maximum Principle by Means of Needle Variations

We propose a proof of the maximum principle for the general Pontryagin type optimal control problem, based on packets of needle variations. The optimal control problem is first reduced to a family of smooth finite-dimensional problems, the arguments of which are the widths of the needles in each packet, then, for each of these problems, the standard Lagrange multipliers rule is applied, and finally, the obtained family of necessary conditions is “compressed” in one universal optimality condition by using the concept of centered family of compacta.

Bounded Lagrange multiplier rules for general nonsmooth problems and application to mathematical programs with equilibrium constraints

In this paper, a general optimization problem is considered to investigate the conditions which ensure the existence of Lagrangian vectors with a norm not greater than a fixed positive number. In addition, the nonemptiness and boundedness of the multiplier sets together with their exact upper bounds is characterized. Moreover, three new constraint qualifications are suggested that each of them follows a degree of boundedness for multiplier vectors. Several examples at the end of the paper indicate that the upper bound for Lagrangian vectors is easily computable using each of our constraint qualifications. One innovation is introducing the so-called bounded Lagrangian constraint qualification which is stated based on the nonemptiness and boundedness of all possible Lagrangian sets. An application of the results for a mathematical program with equilibrium constraints is presented.

A new proof of the Lagrange multiplier rule

We present an elementary self-contained proof for the Lagrange multiplier rule. It does not refer to any preliminary material and it is only based on the observation that a certain limit is positive. At the end of this note, the power of the Lagrange multiplier rule is analyzed.

Generalized Lagrange multiplier rule for non-convex vector optimization problems

In this paper a non-convex vector optimization problem among infinite-dimensional spaces is presented. In particular, a generalized Lagrange multiplier rule is formulated as a necessary and sufficient optimality condition for weakly minimal solutions of a constrained vector optimization problem, without requiring that the ordering cone that defines the inequality constraints has non-empty interior. This paper extends the result of Donato (J. Funct. Analysis261 (2011), 2083–2093) to the general setting of vector optimization by introducing a constraint qualification assumption that involves the Fréchet differentiability of the maps and the tangent cone to the image set. Moreover, the constraint qualification is a necessary and sufficient condition for the Lagrange multiplier rule to hold.

Control of M|M|1|N queue parameters under constraints

An optimal control of the MM1N queue is considered on the basis of the minimum criterion for the expected service time if constraints on the average number of rejected demands and average energy expenditure are present. Two variants of an input flow are studied: with the separation of demands according to priorities and without such a separation. A dual optimization method and a Lagrange multiplier rule are used to synthesize a constrained optimal control. Two iteration procedures, the effectiveness of which is analyzed on the basis of the numerical simulation results, are presented in order to solve numerically a dual problem.

Optimality Conditions in Control Problems for Systems Described by Equations with Monotone Operators

Necessary and sufficient conditions of optimality are proved for some classes of constrained optimization problems with constraints in the form of operator and differential-operator equations. The optimization problems are considered subject to additional functional constraints. The Pontryagin maximum principle and the Lagrange multiplier rule are derived for the relevant problems from the optimality conditions proved in this article.

Boundedness of certain sets of Lagrange multipliers in vector optimization

In this paper, we establish Lagrange multiplier rules in terms of Michel–Penot subdifferential for nonsmooth vector optimization problem. A constraint qualification or regularity condition in terms of Michel–Penot subdifferential is given and under this regularity condition the boundedness of certain sets of Lagrange multipliers are discussed.

The optimal control problem for the coefficients of a parabolic equation under phase constraints

For the optimal control problem on the coefficients of parabolic equations, we study correctness problems for the setting in the weak topology on the space of controls. We show that in this problem, the objective functional is bounded from below, the set of optimal controls is nonempty, and every minimizing sequence weakly converges to the set of optimal controls. We establish a necessary optimality condition in the form of a generalized Lagrange multiplier rule.

A Variational Approach to Lagrange Multipliers

We discuss Lagrange multiplier rules from a variational perspective. This allows us to highlight many of the issues involved and also to illustrate how broadly an abstract version can be applied.

The Pontryagin Maximum Principle for Nonlinear Optimal Control Problems with Infinite Horizon

The famous proof of the Pontryagin maximum principle for control problems on a finite horizon bases on the needle variation technique, as well as the separability concept of cones created by disturbances of the trajectories. In this paper, we follow the approach to state the optimal control problem as an extremal problem in function spaces, and then realizing the Lagrange principle for this extremal problem via the separation technique. The result is the Pontryagin maximum principle as necessary condition for a strong local minimizer in infinite horizon optimal control problems. This approach yields the existence of the adjoint and the validity of the transversality conditions at infinity. To get the Lagrange multipliers rule in this particular context, we introduce a multiple needle variation technique on the infinite horizon.

Noether Theorem for Nonholonomic Systems with Time Delay

The paper focuses on studying the Noether theorem for nonholonomic systems with time delay. Firstly, the differential equations of motion for nonholonomic systems with time delay are established, which is based on the Hamilton principle with time delay and the Lagrange multiplier rules. Secondly, based upon the generalized quasi-symmetric transformations for nonconservative systems with time delay, the Noether theorem for corresponding holonomic systems is given. Finally, we obtain the Noether theorem for the nonholonomic nonconservative systems with time delay. At the end of the paper, an example is given to illustrate the application of the results.