scholarly journals On the regularization of the Lagrange principle and on the construction of the generalized minimizing sequences in convex constrained optimization problems

2020 ◽  
Vol 30 (3) ◽  
pp. 410-428
Author(s):  
M.I. Sumin
Keyword(s):  
Hilbert Space ◽  
Constrained Optimization ◽  
Optimization Problems ◽  
Inequality Constraints ◽  
Functional Inequality ◽  
Minimizing Sequences ◽  
Lagrange Principle ◽  
Strongly Convex ◽  
Regularization Parameters ◽  
Objective Functional

We consider the regularization of the Lagrange principle (LP) in the convex constrained optimization problem with operator constraint-equality in a Hilbert space and with a finite number of functional inequality-constraints. The objective functional of the problem is not, generally speaking, strongly convex. The set of admissible elements of the problem is also embedded into a Hilbert space and is not assumed to be bounded. Obtaining a regularized LP is based on the dual regularization method and involves the use of two regularization parameters and two corresponding matching conditions at the same time. One of the regularization parameters is «responsible» for the regularization of the dual problem, while the other is contained in a strongly convex regularizing addition to the objective functional of the original problem. The main purpose of the regularized LP is the stable generation of generalized minimizing sequences that approximate the exact solution of the problem by function and by constraint, for the purpose of its practical stable solving.

On the regularization of the Lagrange principle and on the construction of the generalized minimizing sequences in convex constrained optimization problems

2020 ◽  
Author(s):  
M. L. Sumin
Keyword(s):  
Constrained Optimization ◽  
Optimization Problems ◽  
Minimizing Sequences ◽  
Lagrange Principle ◽  
Constrained Optimization Problems

Рассматривается регуляризация принципа Лагранжа (ПЛ) в выпуклой задаче условной оптимизации с операторным ограничением-равенством в гильбертовом пространстве и конечным числом функциональных ограничений-неравенств. Целевой функционал задачи не является, вообще говоря, сильно выпуклым, а на множество ее допустимых элементов, которое также принадлежит гильбертову пространству, не накладывается условие ограниченности. Получение регуляризованного ПЛ основано на методе двойственной регуляризации и предполагает использование двух параметров регуляризации и двух соответствующих условий согласования одновременно. Один из регуляризирующих параметров «отвечает» за регуляризацию двойственной задачи, другой же содержится в сильно выпуклом регуляризирующем добавке к целевому функционалу исходной задачи. Основное предназначение регуляризованного ПЛ — устойчивое генерирование обобщенных минимизирующих последовательностей,аппроксимирующих точное решение задачи по функции и по ограничениям, для целей ее непосредственного практического устойчивого решения

Feasible and Optimal Designs of Variable Air Gap Torque Motors

1991 ◽  
Vol 113 (2) ◽  
pp. 241-245 ◽  
Author(s):  
M. C. Leu ◽  
R. A. Aubrecht
Keyword(s):  
Power Consumption ◽  
Constrained Optimization ◽  
Optimization Problems ◽  
Inequality Constraints ◽  
Physical Characteristics ◽  
Air Gap ◽  
Numerical Results ◽  
Optimal Designs ◽  
Constrained Optimization Problems ◽  
Equality And Inequality Constraints

The problems of automating the feasible and optimal designs of variable air gap torque motors are studied. Both are formulated as constrained optimization problems, where equality and inequality constraints are associated with the geometrical and physical characteristics of the device. Numerical results show that feasible designs can be obtained for specified rated torque outputs, and optimal designs can be achieved by reducing the volume or power consumption substantially from the initial designs, without reducing the rated torque output.

A Quadratic Programming Formulation of the Portfolio Selection Model

1962 ◽  
Vol 16 (06) ◽  
pp. 468-476 ◽  
Author(s):  
P. Wegner
Keyword(s):  
Constrained Optimization ◽  
Optimization Problems ◽  
Inequality Constraints ◽  
Lagrangian Multiplier ◽  
Programming Formulation ◽  
Constrained Optimization Problems ◽  
Non Linear Programming ◽  
Programming Techniques ◽  
Portfolio Selection Model ◽  
Derivatives Of

1. Kennedy & Howroyd (1956) have discussed the application of the Lagrangian multiplier technique to actuarial problems. This technique permits the analytical solution of constrained optimization problems, where both the function to be optimized and the constraints must satisfy stringent analytical conditions. In particular, the first derivatives of all functions must exist.When constraints comprise inequality as well as equation restrictions, as is the case in linear and non-linear programming, then the conditions required for the Lagrangian multiplier technique do not hold. It was therefore found necessary to develop a new body of techniques, known as mathematical programming techniques, for the solution of constrained optimization problems involving inequality constraints.

The Rate of Convergence of a NLM Based on F–B NCP for Constrained Optimization Problems Without Strict Complementarity

2015 ◽  
Vol 32 (03) ◽  
pp. 1550012 ◽  
Author(s):  
Suxiang He ◽  
Liwei Zhang ◽  
Jie Zhang
Keyword(s):  
Constrained Optimization ◽  
Rate Of Convergence ◽  
Constraint Qualification ◽  
Optimization Problems ◽  
Augmented Lagrangian Method ◽  
Inequality Constraints ◽  
Constrained Optimization Problems ◽  
Strict Complementarity ◽  
Linear Rate ◽  
Nonlinear Lagrangian

It is well-known that the linear rate of convergence can be established for the classical augmented Lagrangian method for constrained optimization problems without strict complementarity. Whether this result is still valid for other nonlinear Lagrangian methods (NLM) is an interesting problem. This paper proposes a nonlinear Lagrangian function based on Fischer–Burmeister (F–B) nonlinear complimentarity problem (NCP) function for constrained optimization problems. The rate of convergence of this NLM is analyzed under the linear independent constraint qualification and the strong second-order sufficient condition without strict complementarity when subproblems are assumed to be solved exactly and inexactly, respectively. Interestingly, it is demonstrated that the Lagrange multipliers associating with inactive inequality constraints at the local minimum point converge to zeros superlinearly. Several illustrative examples are reported to show the behavior of the NLM.

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