A Filled Function Method Dominated by Filter for Nonlinearly Global Optimization

This work presents a filled function method based on the filter technique for global optimization. Filled function method is one of the effective methods for nonlinear global optimization, since it can effectively find a better minimizer. Filter technique is applied to local optimization methods for its excellent numerical results. In order to optimize the filled function method, the filter method is employed for global optimizations in this method. A new filled function is proposed first, and then the algorithm and its properties are proved. The numerical results are listed at the end

Finding global minimum using filled function method

Filled function method is an optimization method for finding global minimizers. Filled function method is a combination of a local search in findings local solutions as well as global solution. It is basically a construction and eventually the inclusion of an auxiliary function called the filled function into the algorithm. Optimizing the objective function at an initial point will only yield a local minimizer. By using the auxiliary function, the local minimizer is shifted to a new lower basin of the objective function. The shifted point is the new initial solution for the local search to find the next local minimizer, where the function value is lower. The process continued until the global minimizer is achieved. This research used several test functions to examine the effectiveness of the method in finding global solution. The results show that this method works successfully and further research directions are discussed

Global algorithms for nonlinear discrete optimization and discrete-valued optimal control problems

Optimal control problems arise in many applications, such as in economics, finance, process engineering, and robotics. Some optimal control problems involve a control which takes values from a discrete set. These problems are known as discrete-valued optimal control problems. Most practical discrete-valued optimal control problems have multiple local minima and thus require global optimization methods to generate practically useful solutions. Due to the high complexity of these problems, metaheuristic based global optimization techniques are usually required.One of the more recent global optimization tools in the area of discrete optimization is known as the discrete filled function method. The basic idea of the discrete filled function method is as follows. We choose an initial point and then perform a local search to find an initial local minimizer. Then, we construct an auxiliary function, called a discrete filled function, at this local minimizer. By minimizing the filled function, either an improved local minimizer is found or one of the vertices of the constraint set is reached. Otherwise, the parameters of the filled function are adjusted. This process is repeated until no better local minimizer of the corresponding filled function is found. The final local minimizer is then taken as an approximation of the global minimizer.While the main aim of this thesis is to present a new computational methodfor solving discrete-valued optimal control problems, the initial focus is on solvingpurely discrete optimization problems. We identify several discrete filled functionstechniques in the literature and perform a critical review including comprehensive numerical tests. Once the best filled function method is identified, we propose and test several variations of the method with numerical examples.We then consider the task of determining near globally optimal solutions of discrete-valued optimal control problems. The main difficulty in solving the discrete-valued optimal control problems is that the control restraint set is discrete and hence not convex. Conventional computational optimal control techniques are designed for problems in which the control takes values in a connected set, such as an interval, and thus they cannot solve the problem directly. Furthermore, variable switching times are known to cause problems in the implementation of any numerical algorithm due to the variable location of discontinuities in the dynamics. Therefore, such problem cannot be solved using conventional computational approaches. We propose a time scaling transformation to overcome this difficulty, where a new discrete variable representing the switching sequence and a new variable controlling the switching times are introduced. The transformation results in an equivalent mixed discrete optimization problem. The transformed problemis then decomposed into a bi-level optimization problem, which is solved using a combination of an efficient discrete filled function method identified earlier and a computational optimal control technique based on the concept of control parameterization.To demonstrate the applicability of the proposed method, we solve two complex applied engineering problems involving a hybrid power system and a sensor scheduling task, respectively. Computational results indicate that this method is robust, reliable, and efficient. It can successfully identify a near-global solution for these complex applied optimization problems, despite the demonstrated presence of multiple local optima. In addition, we also compare the results obtained with other methods in the literature. Numerical results confirm that the proposed method yields significant improvements over those obtained by other methods

Solving one-dimensional unconstrained global optimization problem using parameter free filled function method

It is generally known that almost all filled function methods for one-dimensional unconstrained global optimization problems have computational weaknesses. This paper introduces a relatively new parameter free filled function, which creates a non-ascending bridge from any local isolated minimizer to other first local isolated minimizer with lower or equal function value. The algorithm’s unprecedented function can be used to determine all extreme and inflection points between the two considered consecutive local isolated minimizers. The proposed method never fails to carry out its job. The results of the several testing examples have shown the capability and efficiency of this algorithm while at the same time, proving that the computational weaknesses of the filled function methods can be overcomed

Global optimum design of uniform FIR filter bank with magnitude constraints

The optimum design of a uniform finite impulse response filter bank can be formulated as a nonlinear semi-infinite optimization problem. However, this optimization problem is nonconvex with infinitely many inequality constraints. In this paper, we propose a new hybrid approach for solving this highly challenging nonlinear, nonconvex semi-infinite optimization problem. In this approach, a gradient-based method is used in conjunction with a filled function method to determine a global minimum of the problem. This new hybrid approach finds an optimal result independent of the initial guess of the solution. The method is applied to some existing examples. The results obtained are superior to those obtained by other existing methods. © 2008 IEEE

Allpass VFD Filter Design

This correspondence proposes a general design for allpass variable fractional delay (VFD) digital filters with minimum weighted integral squared error subject to constraints on maximum error deviation from the desired response. The resulting optimization problem is nonlinear and nonconvex with a nonlinear continuous inequality constraint. Stability of the designed filters are discussed. An effective procedure is proposed for solving the optimization problem. Firstly, a constraint transcription method and a smoothing technique are employed to transform the continuous inequality constraint into one equality constraint. Then, by using the concept of a penalty function, the transformed constraint is incorporated into the cost function to form a new cost function. The nonlinear optimization problem subject to continuous inequality constraints is then approximated by a sequence of unconstraint optimization problems. Finally, a global optimization method using a filled function is employed to solve the unconstraint optimization problem. Design example shows that a trade-off can be achieved between the integral squared error and the maximum error deviation for the design of allpass VFD filters

An augmented filled function for global nonlinear integer optimization

The problem of finding global minima of nonlinear discrete functions arises in many fields of practical matters. In recent years, methods based on discrete filled functions have become popular as ways of solving these sort of problems. However, they rely on the steepest descent method for local searches. Here, we present an approach that does not depend on a particular local optimization method, and a new discrete filled function with the useful property that a good continuous global optimization algorithm applied to it leads to an approximation of the solution of the nonlinear discrete problem (Theorem 4). Numerical results are given showing the efficiency of the new approach.Fil: Di Mauro Aparicio, Juan Pablo. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigación en Ciencias de la Computación. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigación en Ciencias de la Computación; ArgentinaFil: Scolnik, Hugo Daniel. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigación en Ciencias de la Computación. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigación en Ciencias de la Computación; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Computación; Argentin

Towards global solutions of optimal discrete-valued control problems

This paper a new heuristic approach for solving optimal discrete-valued control problem. We illustrate the approach with an existing hybrid power system model. The problem of choosing an operating schedule to minimize generator, battery and switching costs is first posed as a mixed discrete dynamic optimization problem. Then, a discrete filled function method is employed in conjunction with a computational optimal control technique to solve this problem.Computational results indicate that this approach is robust, efficient, and can successfully identify a near-global solution for this complex applied optimization problem despite the presence of multiple local optima

Global optimality conditions and optimization methods for polynomial programming problems and their applications

The polynomial programming problem which has a polynomial objective function, either with no constraints or with polynomial constraints occurs frequently in engineering design, investment science, control theory, network distribution, signal processing and locationallocation contexts. Moreover, the polynomial programming problem is known to be Nondeterministic Polynomial-time hard (NP-hard). The polynomial programming problem has attracted a lot of attention, including quadratic, cubic, homogenous or normal quartic programming problems as special cases. Existing methods for solving polynomial programming problems include algebraic methods and various convex relaxation methods. Especially, among these methods, semidefinite programming (SDP) and sum of squares (SOS) relaxations are very popular. Theoretically, SDP and SOS relaxation methods are very powerful and successful in solving the general polynomial programming problem with a compact feasible region. However, the solvability in practice depends on the size or the degree of the polynomial programming problem and the required accuracy. Hence, solving large scale SDP problems still remains a computational challenge. It is well-known that traditional local optimization methods are designed based on necessary local optimality conditions, i.e., Karush-Kuhn-Tucker (KKT) conditions. Motivated by this, some researchers proposed a necessary global optimality condition for a quadratic programming problem and designed a new local optimization method according to the necessary global optimality condition. In this thesis, we try to apply this idea to cubic and quatic programming problems, and further to general unconstrained and constrained polynomial programming problems. For these polynomial programming problems, we will investigate necessary global optimality conditions and design new local optimization methods according to these conditions. These necessary global optimality conditions are generally stronger than KKT conditions. Hence, the obtained new local minimizers by using the new local optimization methods may improve some KKT points. Our ultimate aim is to design global optimization methods for these polynomial programming problems. We notice that the filled function method is one of the well-known and practical auxiliary function methods used to achieve a global minimizer. In this thesis, we design global optimization methods by combining the new proposed local optimization methods and some auxiliary functions. The numerical examples illustrate the efficiency and stability of the optimization methods. Finally, we discuss some applications for solving some sensor network localization problems and systems of polynomial equations. It is worth mentioning that we apply the idea and the results for polynomial programming problems to nonlinear programming problems (NLP). We provide an optimality condition and design new local optimization methods according to the optimality condition and design global optimization methods for the problem (NLP) by combining the new local optimization methods and an auxiliary function. In order to test the performance of the global optimization methods, we compare them with two other heuristic methods. The results demonstrate our methods outperform the two other algorithms.Doctor of Philosoph

Two-channel linear phase FIR QMF bank minimax design via global nonconvex optimization programming

In this correspondence, a two-channel linear phase finite impulse response (FIR) quadrature mirror filter (QMF) bank minimax design problem is formulated as a nonconvex optimization problem so that a weighted sum of the maximum amplitude distortion of the filter bank, the maximum passband ripple magnitude and the maximum stopband ripple magnitude of the prototype filter is minimized subject to specifications on these performances. A modified filled function method is proposed for finding the global minimum of the nonconvex optimization problem. Computer numerical simulations show that our proposed design method is efficient and effective