Continuous Optimization Method Research Articles

Convex two-level optimization problem

A two-level optimization problem is considered in which the objective functional of the second-level problem is minimized on the solution set of the first-level problem. Convergence of the modified penalty method is established. The main results include a continuous two-level optimization method based on the regularized extremal shifting principle [3, 5, 6]. For a linearly convex problem, two-sided bounds on the approximation by the first-level functional are established in addition to convergence. For a linearly quadratic problem, two-sided bounds on the approximation by the second-level functional are derived. For a linearly quadratic problem with interval constraints, an explicit form of differential inclusions is presented for the implementation of the method.

Continuous Optimization Method for Integrated Land Use/Transportation Models

A basic framework for the sensitivity analysis and continuous optimization of integrated land use/transportation model is proposed in this paper. It is shown that separate sensitivity analysis procedures for a land use model with exogenous travel costs and a transportation network with elastic demand can be combined to get a sensitivity analysis procedure for the integrated model. Sensitivity-analysis-based techniques can then be used for the optimal design of land use plans and transportation policies so that certain social surplus functions are optimized. The proposed method is illustrated with a residential location model with a congestible transportation network sub-model.

Continuous optimization methods for structure alignments

Structural Alignment is an important tool for fold identification of proteins, structural screening on ligand databases, pharmacophore identification and other applications. In the general case, the optimization problem of superimposing two structures is nonsmooth and nonconvex, so that most popular methods are heuristic and do not employ derivative information. Usually, these methods do not admit convergence theories of practical significance. In this work it is shown that the optimization of the superposition of two structures may be addressed using continuous smooth minimization. It is proved that, using a Low Order-Value Optimization approach, the nonsmoothness may be essentially ignored and classical optimization algorithms may be used. Within this context, a Gauss–Newton method is introduced for structural alignments incorporating (or not) transformations (as flexibility) on the structures. Convergence theorems are provided and practical aspects of implementation are described. Numerical experiments suggest that the Gauss–Newton methodology is competitive with state-of-the-art algorithms for protein alignment both in terms of quality and speed. Additional experiments on binding site identification, ligand and cofactor alignments illustrate the generality of this approach. The softwares containing the methods presented here are available at http://www.ime.unicamp.br/∼martinez/lovoalign.

The Cross-Entropy Method for Continuous Multi-Extremal Optimization

In recent years, the cross-entropy method has been successfully applied to a wide range of discrete optimization tasks. In this paper we consider the cross-entropy method in the context of continuous optimization. We demonstrate the effectiveness of the cross-entropy method for solving difficult continuous multi-extremal optimization problems, including those with non-linear constraints.

Derivative-Free Filter Simulated Annealing Method for Constrained Continuous Global Optimization

In this paper, a simulated-annealing-based method called Filter Simulated Annealing (FSA) method is proposed to deal with the constrained global optimization problem. The considered problem is refo...

A new tabu search method for continuous parameter optimization: application to design problems in electromagnetic

AbstractSpecificities of optimal design problem in electromagnetic are emphasized. To overcome the revolting difficulties met with the classical stochastic methods, a new tabu search algorithm for global optimization of multimodal functions with continuous variables is presented and used for solving some of this kind of electromagnetic optimization problems such as Bruhless DC motor, SMES optimization, and die mold press. It is compared favorably with Hu's tabu search and universal tabu search methods. Copyright © 2005 John Wiley & Sons, Ltd.

Design of Steel Frames Using Ant Colony Optimization

A design procedure utilizing an ant colony optimization (ACO) technique is developed for discrete optimization of steel frames. The objective function considered is the total weight (or cost) of the structure subjected to serviceability and strength requirements as specified by the American Institute for Steel Construction (AISC) Load and Resistance Factor Design, 2001. The design of steel frames is mapped into a modified traveling salesman problem (TSP) where the configuration of the TSP network reflects the structural topology, and the resulting length of the TSP tour corresponds to the weight of the frame. The number of potential paths between nodes in the TSP network represents all (or a portion) of the available W-shapes in the AISC database. The resulting frame, mapped into a TSP, is minimized using an ACO algorithm with a penalty function to enforce strength and serviceability constraints. A comparison is presented between the ACO frame designs and designs developed using a genetic algorithm and classical continuous optimization methods.

The enumeration method for selecting optimum switching network structures

In this paper it is shown that traditional continuous optimization methods for multistage switching networks may fail, leading to nonoptimal structures. Then, a simple enumeration method to solve the optimization problem is proposed. The numerical examples show that the parameters of optimum networks can be calculated very fast for any practical network sizes.

Efficient hybrid methods for global continuous optimization based on simulated annealing

We introduce several hybrid methods for global continuous optimization. They combine simulated annealing and a local proximal bundle method. Traditionally, the simplest hybrid of a global and a local solver is to call the local solver after the global one, but this does not necessarily produce good results. Besides, using efficient gradient-based local solvers implies that the hybrid can only be applied to differentiable problems. We show several ways how to integrate the local solver as a genuine part of simulated annealing to enable both efficient and reliable solution processes. When using the proximal bundle method as a local solver, it is possible to solve even nondifferentiable problems. The numerical tests show that the hybridization can improve both the efficiency and the reliability of simulated annealing.

Global optimization method using chaos of discrete gradient dynamics

In this paper, we propose another method for continuous global optimization which makes use of the chaotic property of discrete gradient models as nonlinear dynamical systems. The deterministic randomness generated in chaotic phase improves the global searching ability of the gradient dynamics. Besides, the global optimizer can be found at high accuracy and ef ciency by applying the modi ed fichaotic annealing algorithmfl by us to the discrete gradient dynamics.

Design of Space Trusses Using Ant Colony Optimization

A design procedure utilizing an ant colony optimization (ACO) technique is developed for discrete optimization of space trusses. The objective function considered is the total weight (or cost) of the structure subjected to material and performance constraints in the form of stress and deflection limits. The design of space trusses using discrete variables is transformed into a modified traveling salesman problem (TSP) where the network of the TSP reflects the structural topology and the length of the TSP tour is the weight of the structure. The resulting truss, mapped into a TSP, is minimized using an ACO algorithm. The ACO design procedure uses discrete design variables, an open format for prescribing constraints, a penalty function to enforce design constraints, and allows for multiple loading cases. A comparison is presented between the ACO truss design procedure and designs developed using a genetic algorithm and classical continuous optimization methods.

Survey of multi-objective optimization methods for engineering

A survey of current continuous nonlinear multi-objective optimization (MOO) concepts and methods is presented. It consolidates and relates seemingly different terminology and methods. The methods are divided into three major categories: methods with a priori articulation of preferences, methods with a posteriori articulation of preferences, and methods with no articulation of preferences. Genetic algorithms are surveyed as well. Commentary is provided on three fronts, concerning the advantages and pitfalls of individual methods, the different classes of methods, and the field of MOO as a whole. The Characteristics of the most significant methods are summarized. Conclusions are drawn that reflect often-neglected ideas and applicability to engineering problems. It is found that no single approach is superior. Rather, the selection of a specific method depends on the type of information that is provided in the problem, the user’s preferences, the solution requirements, and the availability of software.

Steepest descent with momentum for quadratic functions is a version of the conjugate gradient method

It is pointed out that the so called momentum method, much used in the neural network literature as an acceleration of the backpropagation method, is a stationary version of the conjugate gradient method. Connections with the continuous optimization method known as heavy ball with friction are also made. In both cases, adaptive (dynamic) choices of the so called learning rate and momentum parameters are obtained using a control Liapunov function analysis of the system.

A Sequential Convexification Method (SCM) for Continuous Global Optimization

A new method for continuous global minimization problems, acronymed SCM, is introduced. This method gives a simple transformation to convert the objective function to an auxiliary function with gradually `fewer' local minimizers. All Local minimizers except a prefixed one of the auxiliary function are in the region where the function value of the objective function is lower than its current minimal value. Based on this method, an algorithm is designed which uses a local optimization method to minimize the auxiliary function to find a local minimizer at which the value of the objective function is lower than its current minimal value. The algorithm converges asymptotically with probability one to a global minimizer of the objective function. Numerical experiments on a set of standard test problems with several problems' dimensions up to 50 show that the algorithm is very efficient compared with other global optimization methods.

Efficient and Adaptive Lagrange-Multiplier Methods for Nonlinear Continuous Global Optimization

Lagrangian methods are popular in solving continuous constrained optimization problems. In this paper, we address three important issues in applying Lagrangian methods to solve optimization problems with inequality constraints. First, we study methods to transform inequality constraints into equality constraints. An existing method, called the slack-variable method, adds a slack variable to each inequality constraint in order to transform it into an equality constraint. Its disadvantage is that when the search trajectory is inside a feasible region, some satisfied constraints may still pose some effect on the Lagrangian function, leading to possible oscillations and divergence when a local minimum lies on the boundary of the feasible region. To overcome this problem, we propose the MaxQ method that carries no effect on satisfied constraints. Hence, minimizing the Lagrangian function in a feasible region always leads to a local minimum of the objective function. We also study some strategies to speed up its convergence. Second, we study methods to improve the convergence speed of Lagrangian methods without affecting the solution quality. This is done by an adaptive-control strategy that dynamically adjusts the relative weights between the objective and the Lagrangian part, leading to better balance between the two and faster convergence. Third, we study a trace-based method to pull the search trajectory from one saddle point to another in a continuous fashion without restarts. This overcomes one of the problems in existing Lagrangian methods that converges only to one saddle point and requires random restarts to look for new saddle points, often missing good saddle points in the vicinity of saddle points already found. Finally, we describe a prototype Novel (Nonlinear Optimization via External Lead) that implements our proposed strategies and present improved solutions in solving a collection of benchmarks.

The Cross-Entropy Method for Combinatorial and Continuous Optimization

We present a new and fast method, called the cross-entropy method, for finding the optimal solution of combinatorial and continuous nonconvex optimization problems with convex bounded domains. To find the optimal solution we solve a sequence of simple auxiliary smooth optimization problems based on Kullback-Leibler cross-entropy, importance sampling, Markov chain and Boltzmann distribution. We use importance sampling as an important ingredient for adaptive adjustment of the temperature in the Boltzmann distribution and use Kullback-Leibler cross-entropy to find the optimal solution. In fact, we use the mode of a unimodal importance sampling distribution, like the mode of beta distribution, as an estimate of the optimal solution for continuous optimization and Markov chains approach for combinatorial optimization. In the later case we show almost surely convergence of our algorithm to the optimal solution. Supporting numerical results for both continuous and combinatorial optimization problems are given as well. Our empirical studies suggest that the cross-entropy method has polynomial in the size of the problem running time complexity.

Scheduling hydro power systems with restricted operating zones and discharge ramping constraints

An optimization-based algorithm is presented for scheduling hydro power systems with restricted operating zones and discharge ramping constraints. Hydro watershed scheduling problems are difficult to solve because many constraints, continuous and discrete, including hydraulic coupling of cascaded reservoirs have to be considered. Restricted or forbidden operating zones as well as minimum generation limits of hydro units result in discontinuous preferred operating regions, and hinder direct applications of efficient continuous optimization methods such as network flow algorithms. Discharge ramping constraints due to navigational, environmental and recreational requirements in a hydro system add another dimension of difficulty since they couple generation or water discharge across time horizon. Integrated consideration of the above constraints is very challenging. The key idea of this paper is to use additional sets of multipliers to relax discontinuous operating region and discharge ramping constraints on individual hydro units so that a two-level optimization structure is formed. The low level consists of a continuous discharge scheduling subproblem determining the generation levels of all units in the entire watershed, and a number of pure integer scheduling subproblems determining the hydro operating states, one for each unit. The discharge subproblem is solved by a network flow algorithm, and the integer scheduling problems are solved by dynamic programming with a small number of states and well-structured transitions.

An approximate algorithm for nonlinear integer programming

This paper presents an approximate algorithm, which is modified from the Filled Function method for continuous global optimization, to solve nonlinear integer programming problems. Unlike the other approximate methods, the algorithm tries to improve a discrete local minimal solution by minimizing a filled function. It is a direct method. Numerical experiments show that this algorithm is efficient.

The genetic algorithm applied to stiffness maximization of laminated plates: review and comparison

The design of laminated structures is highly tailorable owing to the large number of available design variables, thereby requiring an optimization method for effective design. Furthermore, in practice, the design problem translates to a discrete global optimization problem which requires a robust optimization method such as the genetic algorithm. In this paper, the genetic algorithm, based on the real variable coding, is applied to the strain energy minimization of rectangular laminated composite plates. The results for both a point load and uniformly distributed load compare well with those achieved using trajectory methods for continuous global optimization.

Optimal design of steel structures using standard sections

The problem of optimal structural design having linked discrete variables is addressed. For such applications, when a discrete value for a variable is selected, values for other variables linked to it must also be selected from a table. The design of steel structures using available sections is a major application area of such problems. Three strategies that combine a continuous variable optimization method with a genetic algorithm, simulated annealing, and branch and bound method are presented and implemented into a computer program for their numerical evaluation. Three structural design problems are solved to study the performance of the proposed methods. CPU times for solution of the problems with discrete variables are large. Strategies are suggested to reduce these times.