Sufficient conditions for solving linearly constrained separable concave global minimization problems

1993 ◽  
Vol 3 (1) ◽  
pp. 79-94 ◽  
Author(s):  
A. T. Phillips ◽  
J. B. Rosen
Keyword(s):  
Sufficient Conditions ◽  
Global Minimization ◽  
Minimization Problems ◽  
Linearly Constrained

A parallel stochastic method for solving linearly constrained concave global minimization problems

1992 ◽  
Vol 2 (3) ◽  
pp. 243-258 ◽  
Author(s):  
A. T. Phillips ◽  
J. B. Rosen ◽  
M. Van Vliet
Keyword(s):  
Stochastic Method ◽  
Global Minimization ◽  
Minimization Problems ◽  
Linearly Constrained

scholarly journals Randomized sketch descent methods for non-separable linearly constrained optimization

2020 ◽  
Author(s):  
Ion Necoara ◽  
Martin Takáč
Keyword(s):  
Objective Function ◽  
Large Scale ◽  
Convergence Rates ◽  
Optimization Problems ◽  
Sufficient Conditions ◽  
Linear Constraints ◽  
Descent Methods ◽  
Constrained Problems ◽  
Special Cases ◽  
Linearly Constrained

Abstract In this paper we consider large-scale smooth optimization problems with multiple linear coupled constraints. Due to the non-separability of the constraints, arbitrary random sketching would not be guaranteed to work. Thus, we first investigate necessary and sufficient conditions for the sketch sampling to have well-defined algorithms. Based on these sampling conditions we develop new sketch descent methods for solving general smooth linearly constrained problems, in particular, random sketch descent (RSD) and accelerated random sketch descent (A-RSD) methods. To our knowledge, this is the first convergence analysis of RSD algorithms for optimization problems with multiple non-separable linear constraints. For the general case, when the objective function is smooth and non-convex, we prove for the non-accelerated variant sublinear rate in expectation for an appropriate optimality measure. In the smooth convex case, we derive for both algorithms, non-accelerated and A-RSD, sublinear convergence rates in the expected values of the objective function. Additionally, if the objective function satisfies a strong convexity type condition, both algorithms converge linearly in expectation. In special cases, where complexity bounds are known for some particular sketching algorithms, such as coordinate descent methods for optimization problems with a single linear coupled constraint, our theory recovers the best known bounds. Finally, we present several numerical examples to illustrate the performances of our new algorithms.

scholarly journals A Multi-Level Optimization Method for Elastic Constants Identification of Composite Laminates

2019 ◽  
Vol 9 (20) ◽  
pp. 4267
Author(s):  
Chien Yang Huang ◽  
Tai Yan Kam
Keyword(s):  
Optimal Design ◽  
Elastic Constants ◽  
Composite Laminates ◽  
Composite Laminate ◽  
Optimization Problem ◽  
Optimization Method ◽  
Global Minimization ◽  
Minimization Problems ◽  
Design Variables ◽  
Multi Level

A new and effective elastic constants identification technique is presented to extract the elastic constants of a composite laminate subjected to uniaxial tensile testing. The proposed technique consists of a new multi-level optimization method that can solve different types of minimization problems, including the extraction of material constants of composite laminates from given strains. In the identification process, the optimization problem is solved by using a stochastic multi-start dynamic search minimization algorithm at the first level in order to obtain the statistics of the quasi-optimal design variables for a set of randomly generated starting points. The statistics of the quasi-optimal elastic constants obtained at this level are used to determine the reduced feasible region in order to formulate the second-level optimization problem. The second-level optimization problem is then solved using the particle swarm algorithm in order to obtain the statistics of the new quasi-optimal elastic constants. The iteration process between the first and second levels of optimization continues until the standard deviations of the quasi-optimal design variables at any level of optimization are less than the prescribed values. The proposed multi-level optimization method, as well as several existing global optimization algorithms, is used to solve a number of well-known mathematical minimization problems to verify the accuracy of the method. For the adopted numerical examples, it has been shown that the proposed method is more efficient and effective than the adopted global minimization algorithms to produce the exact solutions. The proposed method is then applied to identify four elastic constants of a [0°/±45°]s composite laminate using three strains in 0°, 45°, and 90° directions, respectively, of the composite laminate subjected to uniaxial testing. For comparison purposes, several existing global minimization techniques are also used to solve the elastic constants identification problem. Again, it has been shown that the proposed method is capable of producing more accurate results than the adopted available methods. Finally, experimental data are used to demonstrate the applications of the proposed method.

scholarly journals Global Minimization of Nonsmooth Constrained Global Optimization with Filled Function

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Wei-xiang Wang ◽  
You-lin Shang ◽  
Ying Zhang
Keyword(s):  
Original Problem ◽  
Initial Point ◽  
Solution Algorithm ◽  
Global Minimization ◽  
Second Phase ◽  
Constrained Global Optimization ◽  
Filled Function ◽  
Minimization Problems ◽  
Global Optimizer ◽  
Two Phases

A novel filled function is constructed to locate a global optimizer or an approximate global optimizer of smooth or nonsmooth constrained global minimization problems. The constructed filled function contains only one parameter which can be easily adjusted during the minimization. The theoretical properties of the filled function are discussed and a corresponding solution algorithm is proposed. The solution algorithm comprises two phases: local minimization and filling. The first phase minimizes the original problem and obtains one of its local optimizers, while the second phase minimizes the constructed filled function and identifies a better initial point for the first phase. Some preliminary numerical results are also reported.

Linearly constrained global minimization of functions with concave minorants

1996 ◽  
Vol 88 (3) ◽  
pp. 751-763 ◽  
Author(s):  
R. Horst ◽  
M. Nast
Keyword(s):  
Global Minimization ◽  
Linearly Constrained ◽  
Minimization Of Functions
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