Lagrange Multiplier Rule Research Articles

The Perturbation Method and Regularization of the Lagrange Multiplier Rule in Convex Constrained Optimization Problems

The Perturbation Method and Regularization of the Lagrange Multiplier Rule in Convex Constrained Optimization Problems

Globally proper efficiency of set optimization problems based on the certainly set less order relation

In this paper, we investigate the globally proper efficiency of set optimization problems. Firstly, we use the so-called certainly set less order relation to define a new kind of set order relation. Based on the new set order relation, we introduce the notion of the globally proper efficient solution of the set optimization problem. Secondly, we establish Lagrange multiplier rule of the set optimization problem. Finally, we obtain Lagrangian duality theorems and saddle point theorems. We also give some examples to illustrate our results.

Discrete Lagrange problems with constraints valued in a Lie group

The Lagrange problem is established in the discrete field theory subject to constraints with values in a Lie group. For the admissible sections that satisfy a certain regularity condition, we prove that the critical sections of such problems are the solutions of a canonically unconstrained variational problem associated with the Lagrange problem (discrete Lagrange multiplier rule). This variational problem has a discrete Cartan 1-form, from which a Noether theory of symmetries and a multisymplectic form formula are established. The whole theory is applied to the Euler-Poincaré reduction in the discrete field theory, concluding as an illustration with the remarkable example of the harmonic maps of the discrete plane in the Lie group SO(n).

О роли множителей Лагранжа и двойственности в некорректных задачах на условный экстремум. К 60-летию метода регуляризации Тихонова

The important role of Lagrange multipliers and duality in the theory of ill-posed problems for a constrained extremum is discussed. The central attention is paid to the problem of stable approximate finding of a normal (minimum in norm) solution of the operator equation of the first kind Az=u, z∈D⊆Z, where A:Z→U is a linear bounded operator, u∈U is a given element, D⊆Z is a convex closed set, Z,U are Hilbert spaces. As is known, this problem is classical for the theory of ill-posed problems. We consider two problems equivalent to it (from the point of view of the simultaneous existence of their unique solutions) for a constrained extremum, the first of which is the problem (CE1) with a functional inequality constraint 〖∥z∥〗^2→"min" , 〖∥Az-u∥〗^2≤0, z∈D, and the second is the problem (CE2) with operator equality constraint 〖∥z∥〗^2→"min" , Az=u, z∈D. First of all, we show that Tikhonov’s regularization method can be naturally interpreted as a method of stable approximation of the exact solution by extremals of the Lagrange functional for problem (CE1) with simultaneous construction of a maximizing sequence of Lagrange multipliers in its dual problem. In this case, the Lagrange multiplier is the reciprocal of the regularization parameter in the Tikhonov method. In other words, the convergence theorem of the Tikhonov regularization method is given the form of a statement in the form of duality with respect to the problem (CE1). Next, we discuss the role of Tikhonov stabilization for general convex problems in solving problems for constrained extremum and a stable method based on Tikhonov stabilization of the problem dual to (CE2) for solving the original operator equation, which can be considered as a regularization method for the Lagrange multiplier rule for the problem (CE2). The paper discusses the features of each of the two above mentioned approaches to the regularization of solving the original operator equation.

Optimal control of elliptic obstacle problems with mixed boundary conditions

We study optimal control of an elliptic obstacle problem with mixed boundary conditions whose weak formulation is a nonlinear variational inequality. We put the control on both the boundary and obstacles. Existence of optimal solutions is proved and the necessary conditions of optimality are derived by the Lagrange multiplier rule and approximation techniques.

New Notions of Proper Efficiency in Set Optimization with the Set Criterion

In this paper, we introduce new notions of proper efficiency in the sense of Henig for a set optimization problem by using the set criterion of solution. The relationships between them are studied. Also, we compare these concepts with the homologous ones given by considering the vector criterion. Finally, a Lagrange multiplier rule for Henig proper solutions of a set optimization problem with a cone constraint is obtained under convexity hypotheses. Illustrative examples are also given.

DRDNN: A robust model for time-variant nonlinear optimization under multiple equality and inequality constraints

External disturbances ubiquitously exist in time-variant problems, including the time-variant nonlinear optimization problems. To develop feasible neural network for time-variant nonlinear optimization restricted to multiple equality and inequality constraints is a bottleneck problem due to its high complexity. In this paper, a novel disturbance rejection dynamic neural network (DRDNN) is proposed to handle nonlinear optimization with multiple equality and inequality constraints concerning external disturbances. According to Lagrange multiplier rule and Karush–Kuhn–Tucker condition, the original optimization problem restricted to multiple equality and inequality constraints is firstly transformed into a time-variant dynamic equation set. Then, detailed design process of the DRDNN model is developed accordingly. The proposed DRDNN inherently possesses the effectiveness and robustness by leveraging the time-derivative as well as the time-integration information. Theorems and proofs about stability, convergence property, and robustness against different forms of disturbances are provided. Finally, different examples, comparisons as well as tests substantiate the accuracy, superiority and robustness of DRDNN for nonlinear optimization limited by multiple equality and inequality constraints in disturbed scenarios.

Sharp Lagrange multipliers for set-valued optimization problems

In this paper, we give a comparison among some notions of weak sharp minima introduced in Amahroq et al. [Le matematiche J. 73 (2018) 99–114], Durea and Strugariu [Nonlinear Anal. 73 (2010) 2148–2157] and Zhu et al. [Set-Valued Var. Anal. 20 (2012) 637–666] for set-valued optimization problems. Besides, we establish sharp Lagrange multiplier rules for general constrained set-valued optimization problems involving new scalarization functionals based on the oriented distance function. Moreover, we provide sufficient optimality conditions for the considered problems without any convexity assumptions.

Second-Order Lagrange Multiplier Rules in Multiobjective Optimal Control of Infinite Dimensional Systems Under State Constraints and Mixed Pointwise Constraints

Second-Order Lagrange Multiplier Rules in Multiobjective Optimal Control of Infinite Dimensional Systems Under State Constraints and Mixed Pointwise Constraints

A domain decomposition method for the Navier\u2013Stokes equations with stochastic input

The paper is committed to studying the domain decomposition method for the incompressible Navier–Stokes equations(NSEs) with stochastic input. The stochastic input is represented spectrally by employing orthogonal polynomial functionals from the Askey scheme as trial basis to represent the random space, and the stochastic NSEs system are transformed into deterministic ones via the polynomial chaos expansion. The corresponding deterministic equations are transformed into the constrained optimization problem by minimizing the cost function on the common interface after the whole domain decomposed into two sub-domains. The constrained optimization problems are transformed into unconstrained problems by the Lagrange multiplier rule. A gradient method-based approach to the solutions of domain decomposition problem is proposed to solve the unconstrained optimality system. Finally, one numerical simulation experiment for square cavity flow problem with the stochastic boundary conditions are performed to demonstrate the feasibility and applicability of the gradient method.

Higher-order efficiency conditions for constrained vector equilibrium problems

ABSTRACT In this paper, we investigate some optimality conditions of higher-order for nonsmooth nonconvex vector equilibrium problems with constraints (or without constraints) in terms of the higher-order upper and lower Studniarski derivatives in Banach spaces. The calculus rule of all the data involved in the problem is taken into account. Using the notion of higher-order upper Studniarski derivative with the class of m-stable/m-steady functions, we first provide the higher-order necessary optimality conditions for the local weak efficient solution of vector equilibrium problem without constraints and then we present the higher-order necessary and sufficient optimality conditions for the local strict minimum of order m to such problem. We second obtain the necessary and sufficient optimality conditions for those efficient solutions of vector equilibrium problem with cone and equality constraints through Lagrange multiplier rules in finite-dimensional spaces. Final, the higher-order necessary and sufficient optimality conditions in terms of the higher-order upper and lower Studniarski derivatives for the local weak efficient solution of vector equilibrium problem with set, cone and equality constraints in Banach spaces are also established. Some examples are proposed to demonstrate our findings.

Lagrange multiplier rules for weak approximate Pareto solutions to constrained vector optimization problems with variable ordering structures

ABSTRACT In this paper, we consider the weak approximate solutions to constrained vector optimization problems with variable ordering structures. In terms of abstract subdifferentials, normal cones and coderivatives, we establish Lagrange rules for this kind of solutions.

Second-Order Lagrange Multiplier Rules in Multiobjective Optimal Control of Semilinear Parabolic Equations

We consider multiobjective optimal control problems for semilinear parabolic systems subject to pointwise state constraints, integral state-control constraints and pointwise state-control constraints. In addition, the data of the problems need not be twice Frechet differentiable. Employing the second-order directional derivative (in the sense of Demyanov-Pevnyi) for the involved functions, we establish necessary optimality conditions, via second-order Lagrange multiplier rules of Fritz-John type, for local weak Pareto solutions of the problems.

The linear scalarizations and Lagrange multipliers for vector optimization

Vector optimization is an important part of mathematical programming. Its theory and methods have a promising interdisciplinary research field with many significantapplications. In this survey, we mainly introduce the progress on the generalized convexity of vector-valued maps, alternative theorems, linear scalarization methods and Lagrange multiplier rules. We first introduce a class of generalized convexity for the vector-valued and set-valued maps, which is based on image space analysis, and summarize the relationship among them. Secondly, we introduce the development of the alternative theorems in linear systems to nonlinear systems. For nonlinear systems, we focus on the research of the alternative theorem under convexity or generalized convexity assumptions. Its applications in the linear scalarization and the Lagrange multiplier rules on vector optimization problems are summarized.

О правиле множителей Лагранжа в задачах с ограничениями типа равенства, задаваемые квазидифференцируемыми функциями

In recent years, there has been a steadily growing interest in the study of extremal problems with parameters that do not satisfy the standard smoothness assumptions. This is due to both theoretical needs and important practical applications in economics, technology, physics, and other sciences. Rough objects naturally arise in a several areas of systems analysis, nonlinear mechanics, and control processes. In the theory of extremal problems, the main interest is the behavior of functions in the vicinity of points where a local extremum is attained. The local behavior of nonsmooth functions is described by subgradients, which are analogs of the derivative of differentiable functions. Using the concepts of subdifferential and subgradient F. Clarke proved the Lagrange multiplier rule in mathematical programming problems with constraints of the type of equalities and inequalities defined by locally Lipschitz functions. However, there are subclasses of locally Lipschitz functions, the simplest examples of which show that the necessary conditions for an extremum obtained by F. Clarke are rather crude and do not allow one to discard obviously non-optimal points. Such a subclass of nonsmooth functions is the subspace of quasi-differentiable functions. In this article, using the Eckland variational principle, we obtain the Lagrange multiplier rule in terms of quasi-differentials. It is shown by examples that this condition is stronger than the necessary condition of F. Clarke.

Active Sensing of Robot Arms Based on Zeroing Neural Networks: A Biological-Heuristic Optimization Model

Conventional biological-heuristic solutions via zeroing neural network (ZNN) models have achieved preliminary efficiency on time-dependent nonlinear optimization problems handling. However, the investigation on finding a feasible ZNN model to solve the time-dependent nonlinear optimization problems with both inequality and equality constraints still remains stagnant because of the nonlinearity and complexity. To make new progresses on the ZNN for time-dependent nonlinear optimization problems solving, this paper proposes a biological-heuristic optimization model, i.e., inequality and equality constrained optimization ZNN (IECO-ZNN). Such a proposed IECO-ZNN breaks the conditionality that the solutions via ZNN for solving nonlinear optimization problems can not consider the inequality and equality constraints at the same time. The time-dependent nonlinear optimization problem subject to inequality and equality constraints is skillfully converted to a time-dependent equality system by exploiting the Lagrange multiplier rule. The design process for the IECO-ZNN model is presented together with its new architecture illustrated in details. In addition, the conversion equivalence, global stability as well as exponential convergence property are theoretically proven. Moreover, numerical studies, real-world applications to robot arm active sensing, and comparisons sufficiently verify the effectiveness and superiority of the proposed IECO-ZNN model for the time-dependent nonlinear optimization with inequality and equality constraints.

A recurrent neural network applied to optimal motion control of mobile robots with physical constraints

Conventional solutions, such as the conventional recurrent neural network (CRNN) and gradient recurrent neural network (GRNN), for the motion control of mobile robots in the unified framework of recurrent neural network (RNN) are difficult to simultaneously consider both criteria optimization and physical constraints. The limitation of the RNN solution may lead to the damage of mobile robots for exceeding physical constraints during the task execution. To overcome this limitation, this paper proposes a novel inequality and equality constrained optimization RNN (IECORNN) to handle the motion control of mobile robots. Firstly, the real-time motion control problem with both criteria optimization and physical constraints is skillfully converted to a real-time equality system by leveraging the Lagrange multiplier rule. Then, the detailed design process for the proposed IECORNN is presented together with the neural network architecture developed. Afterward, theoretical analyses on the motion control problem conversion equivalence, global stability, and exponential convergence property are rigorously provided. Finally, two numerical simulation verifications and extensive comparisons with other existing RNNs, e.g., the CRNN and the GRNN, based on the mobile robot for two different path-tracking applications sufficiently demonstrate the effectiveness and superiority of the proposed IECORNN for the real-time motion control of mobile robots with both criteria optimization and physical constraints. This work makes progresses in both theory as well as practice, and fills the vacancy in the unified framework of RNN in motion control of mobile robots.

On Lagrange Multiplier Rules for Set-Valued Optimization Problems in the Sense of Set Criterion

In this paper, we obtain a new scalar representation of set-valued optimization problems by using a criterion of solution introduced by Kuroiwa, this criterion is a weaker version of Pareto efficient solution. Following this scalarization, we establish necessary optimality conditions in terms of Fritz John multipliers as well as Karush-Kuhn-Tucker multipliers. Moreover, we give sufficient optimality conditions.

The problem of Lagrange in discrete field theory

A discrete Lagrange problem in a fibered manifold on a cellular complex is defined as a Lagrangian density and a constraint submanifold in the 1-jets space of the manifold. After defining the concepts of admissible section and infinitesimal admissible variation, the aim of these problems is to find admissible sections that are critical for the action functional of the Lagrangian density with respect to the infinitesimal admissible variations. For admissible sections satisfying a certain regularity condition, we prove that such critical sections are the solutions of an extended unconstrained discrete variational problem canonically associated to the problem of Lagrange (discrete Lagrange multiplier rule). This variational problem has a discrete Cartan 1-form, from which a Noether theory of symmetries and a multisymplectic form formula are established for the discrete Lagrange problems considered. Finally, the whole theory is illustrated with an example of geometric interest: the discrete harmonic 1-forms on the elliptic or hyperbolic discrete plane, for which we obtain two explicit variational integrators: one with initial conditions (Cauchy problem) and other with boundary conditions (Dirichlet problem).

Limit Configurations of Schrödinger Systems Versus Optimal Partition for the Principal Eigenvalue of Elliptic Systems

Abstract We study a Schrödinger system of four equations with linear coupling functions and nonlinear couplings, including the case that the corresponding elliptic operators are indefinite. For any given nonlinear coupling β > 0 {\beta>0} , we first use minimizing sequences on a normalized set to obtain a minimizer, which implies the existence of positive solutions for some linear coupling constants μ β , ν β {\mu_{\beta},\nu_{\beta}} by Lagrange multiplier rules. Then, as β → ∞ {\beta\to\infty} , we prove that the limit configurations to the competing system are segregated in two groups, develop a variant of Almgren’s monotonicity formula to reveal the Lipschitz continuity of the limit profiles and establish a kind of local Pohozaev identity to obtain the extremality conditions. Finally, we study the relation between the limit profiles and the optimal partition for principal eigenvalue of the elliptic system and obtain an optimal partition for principal eigenvalues of elliptic systems.