An active set strategy for solving optimization problems with up to 200,000,000 nonlinear constraints

A method of multiplier is presented for solving optimization problems. For large-scale constraint problems, combining the active set strategy, we use the aggregate function to approximate the max-value function. Only a few of functions are involved at each iteration, so the computation for gradient is significantly reduced. The numerical results show that the method is effective.

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An Active Set Sequential Linearization Algorithm for Nonlinear Design Optimization

13th Design Automation Conference: Volume 1 — Design Methods, Computer Graphics, and Expert Systems ◽

10.1115/detc1987-0001 ◽

1987 ◽

Author(s):

N. Tzannetakis ◽

P. Y. Papalambros

Keyword(s):

Design Optimization ◽

Optimization Problems ◽

Programming Algorithm ◽

Linear Programs ◽

Chemical Process Design ◽

Active Set ◽

Active Set Strategy ◽

Nonlinear Design ◽

Working Set ◽

Sequential Linearization

Abstract Solution of nonlinear design optimization problems via a sequence of linear programs is regaining attention for solving certain model classes, such as in structural design and chemical process design. An active set strategy modification of an algorithm by Palacios-Gomez is presented. A special interior linear programming algorithm with active set strategy is used also for solving the subproblem and generating the working set of the outer iterations. Examples are included.

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On Exchange Methods for Nonlinear Semi-Infinite Programs

Asia Pacific Journal of Operational Research ◽

10.1142/s0217595921500433 ◽

2021 ◽

Author(s):

Liping Zhang ◽

Shouqiang Du

Keyword(s):

Original Problem ◽

Optimization Problems ◽

Active Set ◽

Exchange Method ◽

Active Set Strategy ◽

Mild Conditions ◽

Approximate Problem ◽

Exchange Rule ◽

Number Of Iterations ◽

Finite Number Of Iterations

A new exchange method is presented for semi-infinite optimization problems with polyhedron constraints. The basic idea is to use an active set strategy as exchange rule to construct an approximate problem with finitely many constraints at each iteration. Under mild conditions, we prove that the proposed algorithm terminates in a finite number of iterations and guarantees that the solution of the resulting approximate problem at final iteration converges to the solution of the original problem within arbitrarily given tolerance. Numerical results indicate that the proposed algorithm is efficient and promising.

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Relaxed gradient projection algorithm for constrained node-based shape optimization

Structural and Multidisciplinary Optimization ◽

10.1007/s00158-020-02821-y ◽

2021 ◽

Author(s):

Ihar Antonau ◽

Majid Hojjat ◽

Kai-Uwe Bletzinger

Keyword(s):

Shape Optimization ◽

Optimization Problems ◽

Gradient Projection ◽

Projection Algorithm ◽

Design Parameters ◽

Active Set ◽

Physical Constraints ◽

Original Algorithm ◽

Gradient Projection Algorithm ◽

Non Linear

AbstractIn node-based shape optimization, there are a vast amount of design parameters, and the objectives, as well as the physical constraints, are non-linear in state and design. Robust optimization algorithms are required. The methods of feasible directions are widely used in practical optimization problems and know to be quite robust. A subclass of these methods is the gradient projection method. It is an active-set method, it can be used with equality and non-equality constraints, and it has gained significant popularity for its intuitive implementation. One significant issue around efficiency is that the algorithm may suffer from zigzagging behavior while it follows non-linear design boundaries. In this work, we propose a modification to Rosen’s gradient projection algorithm. It includes the efficient techniques to damp the zigzagging behavior of the original algorithm while following the non-linear design boundaries, thus improving the performance of the method.

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An Active Set Trust-Region Method for Bound-Constrained Optimization

Bulletin of the Iranian Mathematical Society ◽

10.1007/s41980-021-00610-x ◽

2021 ◽

Author(s):

Morteza Kimiaei

Keyword(s):

Constrained Optimization ◽

Optimization Problems ◽

Trust Region Method ◽

Trust Region ◽

Positive Definiteness ◽

Active Set ◽

Sufficient Descent Condition ◽

Critical Measure ◽

Complexity Result ◽

Bound Constrained Optimization

AbstractThis paper discusses an active set trust-region algorithm for bound-constrained optimization problems. A sufficient descent condition is used as a computational measure to identify whether the function value is reduced or not. To get our complexity result, a critical measure is used which is computationally better than the other known critical measures. Under the positive definiteness of approximated Hessian matrices restricted to the subspace of non-active variables, it will be shown that unlimited zigzagging cannot occur. It is shown that our algorithm is competitive in comparison with the state-of-the-art solvers for solving an ill-conditioned bound-constrained least-squares problem.

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Active set strategy of optimized extreme learning machine

Chinese Science Bulletin ◽

10.1007/s11434-014-0512-2 ◽

2014 ◽

Vol 59 (31) ◽

pp. 4152-4160 ◽

Cited By ~ 4

Author(s):

Xiao-Jian Ding ◽

Bao-Fang Chang

Keyword(s):

Extreme Learning Machine ◽

Active Set ◽

Active Set Strategy ◽

Learning Machine

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Nonlinear Optimal Design of Dynamic Mechanical Systems

22nd Biennial Mechanisms Conference: Mechanism Design and Synthesis ◽

10.1115/detc1992-0350 ◽

1992 ◽

Author(s):

T. E. Potter ◽

K. D. Willmert ◽

M. Sathyamoorthy

Keyword(s):

Objective Function ◽

Optimization Problems ◽

Iteration Process ◽

Optimization Method ◽

Linear Constraints ◽

Sum Of Squares ◽

Function Evaluation ◽

Deformation Analysis ◽

Nonlinear Constraints ◽

Quadratic Constraints

Abstract Mechanism path generation problems which use link deformations to improve the design lead to optimization problems involving a nonlinear sum-of-squares objective function subjected to a set of linear and nonlinear constraints. Inclusion of the deformation analysis causes the objective function evaluation to be computationally expensive. An optimization method is presented which requires relatively few objective function evaluations. The algorithm, based on the Gauss method for unconstrained problems, is developed as an extension of the Gauss constrained technique for linear constraints and revises the Gauss nonlinearly constrained method for quadratic constraints. The derivation of the algorithm, using a Lagrange multiplier approach, is based on the Kuhn-Tucker conditions so that when the iteration process terminates, these conditions are automatically satisfied. Although the technique was developed for mechanism problems, it is applicable to any optimization problem having the form of a sum of squares objective function subjected to nonlinear constraints.

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Evolutionary Algorithms for Constrained Parameter Optimization Problems

Evolutionary Computation ◽

10.1162/evco.1996.4.1.1 ◽

1996 ◽

Vol 4 (1) ◽

pp. 1-32 ◽

Cited By ~ 1027

Author(s):

Zbigniew Michalewicz ◽

Marc Schoenauer

Keyword(s):

Nonlinear Programming ◽

Evolutionary Algorithms ◽

Evolutionary Computation ◽

Numerical Optimization ◽

Optimization Problems ◽

Optimization Techniques ◽

Test Cases ◽

Nonlinear Constraints ◽

Nonlinear Programming Problem ◽

General Nonlinear Programming

Evolutionary computation techniques have received a great deal of attention regarding their potential as optimization techniques for complex numerical functions. However, they have not produced a significant breakthrough in the area of nonlinear programming due to the fact that they have not addressed the issue of constraints in a systematic way. Only recently have several methods been proposed for handling nonlinear constraints by evolutionary algorithms for numerical optimization problems; however, these methods have several drawbacks, and the experimental results on many test cases have been disappointing. In this paper we (1) discuss difficulties connected with solving the general nonlinear programming problem; (2) survey several approaches that have emerged in the evolutionary computation community; and (3) provide a set of 11 interesting test cases that may serve as a handy reference for future methods.

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Implementation of Semi-Heuristic Reasoning for Boundedness Analysis of Design Optimization Models

13th Design Automation Conference: Volume 1 — Design Methods, Computer Graphics, and Expert Systems ◽

10.1115/detc1987-0008 ◽

1987 ◽

Author(s):

J. R. J. Rao ◽

P. Y. Papalambros

Keyword(s):

Optimal Design ◽

Heuristic Search ◽

Optimization Models ◽

Programming Environment ◽

Initial Model ◽

Active Set ◽

Design Models ◽

Active Set Strategy ◽

Reasoning System ◽

Global Boundedness

Abstract A production system performing global boundedness analysis of optimal design models has been implemented in the OPS5 programming environment. The system receives as input an initial model monotonicity table and derives global facts about boundedness and constraint activity using monotonicity principles. Additional facts may be discovered by heuristic search of implicit elimination sequences that examine boundedness of reduced models with active constraints eliminated. The global facts generated automatically by this reasoning system can be used either for a global solution, or for a combined local-global active set strategy.

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