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.

scholarly journals An Augmented Lagrangian Method for Cardinality-Constrained Optimization Problems

2021 ◽  
Author(s):  
Christian Kanzow ◽  
Andreas B. Raharja ◽  
Alexandra Schwartz
Keyword(s):  
Constrained Optimization ◽  
Numerical Experiments ◽  
Penalty Method ◽  
Augmented Lagrangian ◽  
Constraint Qualification ◽  
Optimization Problems ◽  
Augmented Lagrangian Method ◽  
Constrained Optimization Problems ◽  
Strongly Stationary ◽  
Type Constraint

AbstractA reformulation of cardinality-constrained optimization problems into continuous nonlinear optimization problems with an orthogonality-type constraint has gained some popularity during the last few years. Due to the special structure of the constraints, the reformulation violates many standard assumptions and therefore is often solved using specialized algorithms. In contrast to this, we investigate the viability of using a standard safeguarded multiplier penalty method without any problem-tailored modifications to solve the reformulated problem. We prove global convergence towards an (essentially strongly) stationary point under a suitable problem-tailored quasinormality constraint qualification. Numerical experiments illustrating the performance of the method in comparison to regularization-based approaches are provided.

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.

scholarly journals Sequential optimality conditions for cardinality-constrained optimization problems with applications

2021 ◽  
Author(s):  
Christian Kanzow ◽  
Andreas B. Raharja ◽  
Alexandra Schwartz
Keyword(s):  
Constrained Optimization ◽  
Optimality Conditions ◽  
Optimality Condition ◽  
Constraint Qualification ◽  
Optimization Problems ◽  
Local Minimizer ◽  
Constrained Optimization Problems ◽  
New Approach ◽  
Mathematical Programs ◽  
Sequential Optimality Conditions

AbstractRecently, a new approach to tackle cardinality-constrained optimization problems based on a continuous reformulation of the problem was proposed. Following this approach, we derive a problem-tailored sequential optimality condition, which is satisfied at every local minimizer without requiring any constraint qualification. We relate this condition to an existing M-type stationary concept by introducing a weak sequential constraint qualification based on a cone-continuity property. Finally, we present two algorithmic applications: We improve existing results for a known regularization method by proving that it generates limit points satisfying the aforementioned optimality conditions even if the subproblems are only solved inexactly. And we show that, under a suitable Kurdyka–Łojasiewicz-type assumption, any limit point of a standard (safeguarded) multiplier penalty method applied directly to the reformulated problem also satisfies the optimality condition. These results are stronger than corresponding ones known for the related class of mathematical programs with complementarity constraints.

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.

scholarly journals A parallelizable augmented Lagrangian method applied to large-scale non-convex-constrained optimization problems

2018 ◽  
Vol 175 (1-2) ◽  
pp. 503-536 ◽  
Author(s):  
Natashia Boland ◽  
Jeffrey Christiansen ◽  
Brian Dandurand ◽  
Andrew Eberhard ◽  
Fabricio Oliveira
Keyword(s):  
Constrained Optimization ◽  
Large Scale ◽  
Augmented Lagrangian ◽  
Optimization Problems ◽  
Augmented Lagrangian Method ◽  
Lagrangian Method ◽  
Constrained Optimization Problems
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