相关论文
2013 - Soft Computing
A prediction-based adaptive grouping differential evolution algorithm for constrained numerical optimization

In this paper, a new adaptive grouping differential evolution (AGDE) algorithm is proposed to improve the optimization performance by implementing a prediction strategy of the constraints for constrained optimization problems. It is unnecessary to calculate the constraint values when dealing with the constraints in this method. The constraints are handled after a simple prediction according to the Lipschitz condition. When the constraints are very complex, the load arisen from the calculation of the constraint values is reduced dramatically and the feasibility of the solutions remains with great probability. In AGDE algorithm, the population is dynamically grouped to three subpopulations with respective newly-designed mutation strategy. Meanwhile, the mutation factor and crossover probability are adopted associated with the evolutionary process according to the information of the entire population. Both of the above improvements not only increase the diversity of population and speed up the convergence, but also reduce the complexity of the parameter selection. Four sets of comparative experiments are carried out to evaluate the feasibility and effectiveness of the proposed method that deals with the constraints.

2017 - IEEE Transactions on Industrial Electronics
Synchronous Optimal Modulation: Frequency or Time Domain, Global or Constrained Optimization? 40 Years to the First Formulation

Alternative approaches to total harmonic distortion (THD) minimization comprise frequency or time domain THD calculations and global or constrained optimization problem formulation. Time domain constrained optimization formulation suggested in 1977 that seems most adequate from computational efficiency and robustness does not attract sufficient attention it deserves from the modern researchers. Time domain current THD/harmonic loss constrained optimization is illustrated by single-phase inverter examples. New results on multilevel THD optimization for variable voltage levels are presented.

2013 - International Journal for Numerical Methods in Engineering
Topology optimization by a time‐dependent diffusion equation

SUMMARY Most topology optimization problems are formulated as constrained optimization problems; thus, mathematical programming has been the mainstream. On the other hand, solving topology optimization problems using time evolution equations, seen in the level set-based and the phase field-based methods, is yet another approach. One issue is the treatment of multiple constraints, which is difficult to incorporate within time evolution equations. Another issue is the extra re-initialization steps that interrupt the time integration from time to time. This paper proposes a way to describe, using a Heaviside projection-based representation, a time-dependent diffusion equation that addresses these two issues. The constraints are treated using a modified augmented Lagrangian approach in which the Lagrange multipliers are updated by simple ordinary differential equations. The proposed method is easy to implement using a high-level finite element code. Also, it is very practical in the sense that one can fully utilize the existing framework of the code: GUI, parallelized solvers, animations, data imports/exports, and so on. The effectiveness of the proposed method is demonstrated through numerical examples in both the planar and spatial cases. Copyright © 2012 John Wiley & Sons, Ltd.

2015 - Comput. Chem. Eng.
Some efficient approaches for multi-objective constrained optimization of computationally expensive black-box model problems

Abstract Multi-objective constrained optimization problems which arise in many engineering fields often involve computationally expensive black-box model simulators of industrial processes which have to be solved with limited computational time budget, and hence limited number of simulator calls. This paper proposes two heuristic approaches aiming to build proxy problem models, solvable by computationally efficient optimization methods, in order to quickly provide a sufficiently accurate approximation of the Pareto front. The first approach builds a multi-objective mixed-integer linear programming (MO-MILP) surrogate model of the optimization problem relying on piece-wise linear approximations of objectives and constraints obtained through brute-force sensitivity computation. The second approach builds a multi-objective nonlinear programming (MO-NLP) surrogate model using curve fitting of objectives and constraints. In both approaches the desired number of approximated solutions of the Pareto front are generated by applying the ɛ-constraint method to the multi-objective surrogate problems. The proposed approaches are tested for the cost vs. life cycle assessment (LCA)-based environmental optimization of drinking water production plants. The results obtained with both approaches show that a good quality approximation of Pareto front can be obtained with a significantly smaller computational time than with a state-of-the-art metaheuristic algorithm.

2018 - SIAM J. Control. Optim.
An Augmented Lagrangian Method for Optimization Problems in Banach Spaces

We propose a variant of the classical augmented Lagrangian method for constrained optimization problems in Banach spaces. Our theoretical framework does not require any convexity or second-order assumptions, and it allows the treatment of inequality constraints with infinite-dimensional image space. Moreover, we discuss the convergence properties of our algorithm with regard to feasibility, global optimality, and KKT conditions. Some numerical results are given to illustrate the practical viability of the method.

2013 - 2013 IEEE Congress on Evolutionary Computation
An evolutionary algorithm based pattern search approach for constrained optimization

Constrained optimization is one of the popular research areas since constraints are usually present in most real world optimization problems. The purpose of this work is to develop a gradient free constrained global optimization methodology to solve this type of problems. In the methodology proposed, the single objective constrained optimization problem is solved using a Multi-Objective Evolutionary Algorithm (MOEA) by considering two objectives simultaneously, the original objective function and a measure of constraint violation. The MOEA incorporates a penalty function where the penalty parameter is estimated adaptively. The use of penalty function method will enable to further improve the current best solution by decreasing the level of constraint violation, which is made using a gradient free local search method. The performance of the proposed methodology was assessed on a set of benchmark test problems. The results obtained allowed to conclude that the present approach is competitive when compared with other methods available.

1986
Variants of the reduced Newton method for nonlinear equality constrained optimization problems

Following an elementary substitution approach efficient quadratically convergent algorithms for nonlinear equality constrained optimization problems are derived, which reduce the dimension of the problem and allow to exploit special structures of the constraints. The algorithms obtained are compared with those of the Lagrange-Newton technique.

2012 - Computational Optimization and Applications
A genetic algorithm based augmented Lagrangian method for constrained optimization

Among the penalty based approaches for constrained optimization, augmented Lagrangian (AL) methods are better in at least three ways: (i) they have theoretical convergence properties, (ii) they distort the original objective function minimally, thereby providing a better function landscape for search, and (iii) they can result in computing optimal Lagrange multiplier for each constraint as a by-product. Instead of keeping a constant penalty parameter throughout the optimization process, these algorithms update the parameters (called multipliers) adaptively so that the corresponding penalized function dynamically changes its optimum from the unconstrained minimum point to the constrained minimum point with iterations. However, the flip side of these algorithms is that the overall algorithm requires a serial application of a number of unconstrained optimization tasks, a process that is usually time-consuming and tend to be computationally expensive. In this paper, we devise a genetic algorithm based parameter update strategy to a particular AL method. The proposed strategy updates critical parameters in an adaptive manner based on population statistics. Occasionally, a classical optimization method is used to improve the GA-obtained solution, thereby providing the resulting hybrid procedure its theoretical convergence property. The GAAL method is applied to a number of constrained test problems taken from the evolutionary algorithms (EAs) literature. The number of function evaluations required by GAAL in most problems is found to be smaller than that needed by a number of existing evolutionary based constraint handling methods. GAAL method is found to be accurate, computationally fast, and reliable over multiple runs. Besides solving the problems, the proposed GAAL method is also able to find the optimal Lagrange multiplier associated with each constraint for the test problems as an added benefit—a matter that is important for a sensitivity analysis of the obtained optimized solution, but has not yet been paid adequate attention in the past evolutionary constrained optimization studies.

2001 - NIPS
Prodding the ROC Curve: Constrained Optimization of Classifier Performance

When designing a two-alternative classifier, one ordinarily aims to maximize the classifier's ability to discriminate between members of the two classes. We describe a situation in a real-world business application of machine-learning prediction in which an additional constraint is placed on the nature of the solution: that the classifier achieve a specified correct acceptance or correct rejection rate (i.e., that it achieve a fixed accuracy on members of one class or the other). Our domain is predicting churn in the telecommunications industry. Churn refers to customers who switch from one service provider to another. We propose four algorithms for training a classifier subject to this domain constraint, and present results showing that each algorithm yields a reliable improvement in performance. Although the improvement is modest in magnitude, it is nonetheless impressive given the difficulty of the problem and the financial return that it achieves to the service provider.

2008 - IEEE Transactions on Medical Imaging
A Fast Nonrigid Image Registration With Constraints on the Jacobian Using Large Scale Constrained Optimization

This paper presents a new nonrigid monomodality image registration algorithm based on B-splines. The deformation is described by a cubic B-spline field and found by minimizing the energy between a reference image and a deformed version of a floating image. To penalize noninvertible transformation, we propose two different constraints on the Jacobian of the transformation and its derivatives. The problem is modeled by an inequality constrained optimization problem which is efficiently solved by a combination of the multipliers method and the L-BFGS algorithm to handle the large number of variables and constraints of the registration of 3-D images. Numerical experiments are presented on magnetic resonance images using synthetic deformations and atlas based segmentation.

2007 - IEEE Signal Processing Letters
A New Formulation for Empirical Mode Decomposition Based on Constrained Optimization

The empirical mode decomposition (EMD) is an algorithmic construction that aims at decomposing a signal into several modes called intrinsic mode functions. In this letter, we present a new approach for the EMD based on the direct construction of the mean envelope of the signal. The definition of the mean envelope is achieved through the resolution of a quadratic programming problem with equality and inequality constraints. Some numerical experiments conclude this letter, and comparisons are carried out with the classical EMD.

2002 - Engineering Optimization
A socio-behavioural simulation model for engineering design optimization

This paper proposes a method for solving single objective constrained optimization problems by way of a socio-behavioural simulation model. The essence of the methodology is derived from the concept that the behaviour of an individual changes and improves due to social interaction with the society leaders. Leaders are identified after all individuals of a society are Pareto ranked according to constraint satisfaction. At the higher end, leaders of all societies interact among themselves for the overall improvement of the societies. Such overall improvement of individual societies leads to a better civilization. Four well-studied single objective constrained optimization problems have been solved to show the efficacy of the proposed methodology.

2000 - SIAM J. Matrix Anal. Appl.
Upper Perturbation Bounds of Weighted Projections, Weighted and Constrained Least Squares Problems

At each iteration step for solving mathematical programming and constrained optimization problems by using interior-point methods, one often needs to solve the weighted least squares (WLS) problem $\min_{x \in R^n}\| W^{1 \over 2}(Ax-b)\|$, or the weighted and constrained least squares (WLSE) problem $\min_{ x \in R^n}\|W^{1\over 2}(Kx-g)\|$ subject to Lx=h, where W= diag(w1, . . . , wl)>0 in which some $w_i\rightarrow +\infty $ and some $w_i\rightarrow 0$. \par In this paper we will derive upper perturbation bounds of weighted projections associated with the WLS and WLSE problems when W ranges over the set $\cal D$ of positive diagonal matrices. We then apply these bounds to deduce upper perturbation bounds of solutions of WLS and WLSE problems when W ranges over $\cal D$. We also extend the estimates to the cases when W ranges over a subset of real symmetric positive semidefinite matrices.

1997 - SIAM J. Optim.
A Global Convergence Theory for General Trust-Region-Based Algorithms for Equality Constrained Optimization

This work presents a global convergence theory for a broad class of trust-region algorithms for the smooth nonlinear programming problem with equality constraints. The main result generalizes Powell's 1975 result for unconstrained trust-region algorithms. The trial step is characterized by very mild conditions on its normal and tangential components. The normal component need not be computed accurately. The theory requires a quasi-normal component to satisfy a fraction of Cauchy decrease condition on the quadratic model of the linearized constraints. The tangential component then must satisfy a fraction of Cauchy decrease condition on a quadratic model of the Lagrangian function in the translated tangent space of the constraints determined by the quasi-normal component. Estimates of the Lagrange multipliers and the Hessians are assumed only to be bounded. The other main characteristic of this class of algorithms is that the step is evaluated by using the augmented Lagrangian as a merit function with the penalty parameter updated using the El-Alem scheme. The properties of the step and the way that the penalty parameter is chosen are sufficient to establish global convergence. As an example, an algorithm is presented that can be viewed as a generalization of the Steihaug--Toint dogleg algorithm for the unconstrained case. It is based on a quadratic programming algorithm that uses a step in a quasi-normal direction to the tangent space of the constraints and then takes feasible conjugate reduced-gradient steps to solve the reduced quadratic program. This algorithm should cope quite well with large problems for which effective preconditioners are known.

1982
On the Local Convergence of Quasi-Newton Methods for Constrained Optimization

We consider the application of a general class of quasi-Newton methods to the solution of the classical equality constrained nonlinear optimization problem. Specifically, we develop necessary and sufficient conditions for the Q-superlinear convergence of such methods and present a companion linear convergence theorem. The essential conditions relate to the manner in which the Hessian of the Lagrangian function is approximated.

2019 - Appl. Soft Comput.
A survey on projection neural networks and their applications

Abstract Constrained optimization problems arise in numerous scientific and engineering applications, and many papers on the online solution of constrained optimization problems using projection neural networks have been published in the literature. The purpose of this paper is to provide a comprehensive review of the research on projection neural networks for solving various constrained optimizations as well as their applications. Since convergence and stability are important for projection neural networks, theoretical results of projection neural networks are reviewed in detail. In addition, various applications of projection neural networks, e.g., the motion generation of redundant robot manipulators, coordination control of multiple robots with limited communications, generation of winner-take-all strategy, model predictive control and WSN localizations, are discussed and compared. Concluding remarks and future directions of projection neural networks as well as their applications are provided.

2000 - SIAM J. Optim.
A Nonlinear Lagrangian Approach to Constrained Optimization Problems

In this paper we study nonlinear Lagrangian functions for constrained optimization problems which are, in general, nonlinear with respect to the objective function. We establish an equivalence between two types of zero duality gap properties, which are described using augmented Lagrangian dual functions and nonlinear Lagrangian dual functions, respectively. Furthermore, we show the existence of a path of optimal solutions generated by nonlinear Lagrangian problems and show its convergence toward the optimal set of the original problem. We analyze the convergence of several classes of nonlinear Lagrangian problems in terms of their first and second order necessary optimality conditions.

2009 - IEEE Transactions on Antennas and Propagation
Beamforming in the Presence of Mutual Coupling Based on Constrained Particle Swarm Optimization

Two approaches towards beamforming based on constrained particle swarm optimization (PSO) are presented. One approach relies on the penalty function method, for which a new mathematical expression to select the penalty factors is derived. The other method uses a new PSO strategy for solving constrained optimization problems. To maximize performance of the beamforming procedure, the optimal system parameters for the PSO algorithm are derived. Mutual coupling and platform effects are fully accounted for by using the measured active element radiation patterns. The new PSO strategy is compared with a standard genetic algorithm. Using the measured radiation patterns of a seven-element antenna array, a real-life example is presented that demonstrates the possibilities of the new approach.

2008 - GECCO '08
Search space reduction technique for constrained optimization with tiny feasible space

The hurdles in solving Constrained Optimization Problems (COP) arise from the challenge of searching a huge variable space in order to locate feasible points with acceptable solution quality. It becomes even more challenging when the feasible space is very tiny compare to the search space. Usually, the quality of the initial solutions influences the performance of the algorithm in solving such problems. In this paper, we discuss an Evolutionary Agent System (EAS) for solving COPs. In EAS, we treat each individual in the population as an agent. To enhance the performance of EAS for solving COPs with tiny feasible space, we propose a Search Space Reduction Technique (SSRT) as an initial step of our algorithm. SSRT directs the selected infeasible agents in the initial population to move towards the feasible space. The performance of the proposed algorithm is tested on a number of test problems and a real world case problem. The experimental results show that SSRT not only improves the solution quality but also speed up the processing time of the algorithm.

2009 - IEEE Transactions on Systems, Man, and Cybernetics - Part A: Systems and Humans
An Adaptive Penalty Formulation for Constrained Evolutionary Optimization

This paper proposes an adaptive penalty function for solving constrained optimization problems using genetic algorithms. The proposed method aims to exploit infeasible individuals with low objective value and low constraint violation. The number of feasible individuals in the population is used to guide the search process either toward finding more feasible individuals or searching for the optimum solution. The proposed method is simple to implement and does not need any parameter tuning. The performance of the algorithm is tested on 22 benchmark functions in the literature. The results show that the proposed approach is able to find very good solutions comparable to the chosen state-of-the-art designs. Furthermore, it is able to find feasible solutions in every run for all of the benchmark functions tested.

SNOPT: An SQP Algorithm for Large-Scale Constrained Optimization

Michael A. Saunders
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Philip E. Gill
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Walter Murray
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P. Gill
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W. Murray
|
M. Saunders

Abstract:Sequential quadratic programming (SQP) methods have proved highly effective for solving constrained optimization problems with smooth nonlinear functions in the objective and constraints. Here we consider problems with general inequality constraints (linear and nonlinear). We assume that first derivatives are available and that the constraint gradients are sparse. We discuss an SQP algorithm that uses a smooth augmented Lagrangian merit function and makes explicit provision for infeasibility in the original problem and the QP subproblems. SNOPT is a particular implementation that makes use of a semidefinite QP solver. It is based on a limited-memory quasi-Newton approximation to the Hessian of the Lagrangian and uses a reduced-Hessian algorithm (SQOPT) for solving the QP subproblems. It is designed for problems with many thousands of constraints and variables but a moderate number of degrees of freedom (say, up to 2000). An important application is to trajectory optimization in the aerospace industry. Numerical results are given for most problems in the CUTE and COPS test collections (about 900 examples).

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