Global optimization of truss topology with discrete bar areas—Part I: theory of relaxed problems

This two part paper considers the classical problem of finding a truss design with minimal compliance subject to a given external force and a volume bound. The design variables describe the cross-section areas of the bars. While this problem is well-studied for continuous bar areas, we treat here the case of discrete areas. This problem is of major practical relevance if the truss must be built from pre-produced bars with given areas. As a special case, we consider the design problem for a single bar area, i.e., a 0/1-problem. In contrast to heuristic methods considered in other approaches, Part I of the paper together with Part II present an algorithmic framework for the calculation of a global optimizer of the underlying large-scaled mixed integer design problem. This framework is given by a convergent branch-and-bound algorithm which is based on solving a sequence of nonconvex continuous relaxations. The main issue of the paper and of the approach lies in the fact that the relaxed nonlinear optimization problem can be formulated as a quadratic program (QP). Here the paper generalizes and extends the available theory from the literature. Although the Hessian of this QP is indefinite, it is possible to circumvent the non-convexity and to calculate global optimizers. Moreover, the QPs to be treated in the branch-and-bound search tree differ from each other just in the objective function. In Part I we give an introduction to the problem and collect all theory and related proofs for the treatment of the original problem formulation and the continuous relaxed problems. The implementation details and convergence proof of the branch-and-bound methodology and the large-scale numerical examples are presented in Part II.     

Beyond Convex? Global Optimization is Feasible Only for Convex Objective Functions: A Theorem

It is known that there are feasible algorithms for minimizing convex functions, and that for general functions, global minimization is a difficult (NP-hard) problem. It is reasonable to ask whether there exists a class of functions that is larger than the class of all convex functions for which we can still solve the corresponding minimization problems feasibly. In this paper, we prove, in essence, that no such more general class exists. In other words, we prove that global optimization is always feasible only for convex objective functions.     

Additive Scaling and the DIRECT Algorithm

In this paper we show that the convergence behavior of the DIviding RECTangles (DIRECT) algorithm is sensitive to additive scaling of the objective function. We illustrate this problem with a computation and show how the algorithm can be modified to eliminate this sensitivity.     

Solving the ordered one-median problem in the plane

In this paper, we propose a general approach solution method for the single facility ordered median problem in the plane. All types of weights (non-negative, non-positive, and mixed) are considered. The big triangle small triangle approach is used for the solution. Rigorous and heuristic algorithms are proposed and extensively tested on eight different problems with excellent results.     

Tabu search and GRASP for the maximum diversity problem

In this paper, we develop new heuristic procedures for the maximum diversity problem (MDP). This NP-hard problem has a significant number of practical applications such as environmental balance, telecommunication services or genetic engineering. The proposed algorithm is based on the tabu search methodology and incorporates memory structures for both construction and improvement. Although proposed in seminal tabu search papers, memory-based constructions have often been implemented in naïve ways that disregard important elements of the fundamental tabu search proposals. We will compare our tabu search construction with a memory-less design and with previous algorithms recently developed for this problem. The constructive method can be coupled with a local search procedure or a short-term tabu search for improved outcomes. Extensive computational experiments with medium and large instances show that the proposed procedure outperforms the best heuristics reported in the literature within short computational times.     

Hidden Convex Minimization

A class of nonconvex minimization problems can be classified as hidden convex minimization problems. A nonconvex minimization problem is called a hidden convex minimization problem if there exists an equivalent transformation such that the equivalent transformation of it is a convex minimization problem. Sufficient conditions that are independent of transformations are derived in this paper for identifying such a class of seemingly nonconvex minimization problems that are equivalent to convex minimization problems. Thus, a global optimality can be achieved for this class of hidden convex optimization problems by using local search methods. The results presented in this paper extend the reach of convex minimization by identifying its equivalent with a nonconvex representation.     

Finding discrete global minima with a filled function for integer programming

The filled function method is an approach to find the global minimum of multidimensional functions. This paper proposes a new definition of the filled function for integer programming problem. A filled function which satisfies this definition is presented. Furthermore, we discuss the properties of the filled function and design a new filled function algorithm. Numerical experiments on several test problems with up to 50 integer variables have demonstrated the applicability and efficiency of the proposed method.     

On Convergence of a P-Algorithm Based on a Statistical Model of Continuously Differentiable Functions

This paper is a study of the one-dimensional global optimization problem for continuously differentiable functions. We propose a variant of the so-called P-algorithm, originally proposed for a Wiener process model of an unknown objective function. The original algorithm has proven to be quite effective for global search, though it is not efficient for the local component of the optimization search if the objective function is smooth near the global minimizer. In this paper we construct a P-algorithm for a stochastic model of continuously differentiable functions, namely the once-integrated Wiener process. This process is continuously differentiable, but nowhere does it have a second derivative. We prove convergence properties of the algorithm.     

Minimal Ellipsoid Circumscribing a Polytope Defined by a System of Linear Inequalities

In this paper, we will propose algorithms for calculating a minimal ellipsoid circumscribing a polytope defined by a system of linear inequalities. If we know all vertices of the polytope and its cardinality is not very large, we can solve the problem in an efficient manner by a number of existent algorithms. However, when the polytope is defined by linear inequalities, these algorithms may not work since the cardinality of vertices may be huge. Based on a fact that vertices determining an ellipsoid are only a fraction of these vertices, we propose algorithms which iteratively calculate an ellipsoid which covers a subset of vertices. Numerical experiment shows that these algorithms perform well for polytopes of dimension up to seven.     

Optimal and sub-optimal stopping rules for the Multistart algorithm in global optimization

In this paper the problem of stopping the Multistart algorithm for global optimization is considered. The algorithm consists of repeatedly performing local searches from randomly generated starting points. The crucial point in this algorithmic scheme is the development of a stopping criterion; the approach analyzed in this paper consists in stopping the sequential sampling as soon as a measure of the trade-off between the cost of further local searches is greater than the expected benefit, i.e. the possibility of discovering a better optimum.Stopping rules are thoroughly investigated both from a theoretical point of view and from a computational one via extensive simulation. This latter clearly shows that the simple1-step look ahead rule may achieve surprisingly good results in terms of computational cost vs. final accuracy.The research of the second author was partially supported by Progetto MPI 40% Metodi di Ottimizzazione per le Decisioni.     

Location with acceleration–deceleration distance

In this paper we investigate a model where travel time is not necessarily proportional to the distance. Every trip starts at speed zero, then the vehicle accelerates to a cruising speed, stays at the cruising speed for a portion of the trip and then decelerates back to a speed of zero. We define a time equivalent distance which is equal to the travel time multiplied by the cruising speed. This time equivalent distance is referred to as the acceleration–deceleration (A–D) distance. We prove that every demand point is a local minimum for the Weber problem defined by travel time rather than distance. We propose a heuristic approach employing the generalized Weiszfeld algorithm and an optimal approach applying the Big Triangle Small Triangle global optimization method. These two approaches are very efficient and problems of 10,000 demand points are solved in about 0.015 seconds by the generalized Weiszfeld algorithm and in about 1 minute by the BTST technique. When the generalized Weiszfeld algorithm was repeated 1000 times, the optimal solution was found at least once for all test problems.     

A study on concave optimization via canonical dual function

In this study we find a global minimizer of a concave function over a sphere. By introducing a differential equation, we obtain the invariant characteristics for a given optimization problem by constructing a canonical dual function. We present two theorems concerning the global optimality of an extrema of the optimization problem.     

Optimization of Molecular Similarity Index with Applications to Biomolecules

Molecular similarity index measures the similarity between two molecules. Computing the optimal similarity index is a hard global optimization problem. Since the objective function value is very hard to compute and its gradient vector is usually not available, previous research has been based on non-gradient algorithms such as random search and the simplex method. In a recent paper, McMahon and King introduced a Gaussian approximation so that both the function value and the gradient vector can be computed analytically. They then proposed a steepest descent algorithm for computing the optimal similarity index of small molecules. In this paper, we consider a similar problem. Instead of computing atom-based derivatives, we directly compute the derivatives with respect to the six free variables describing the relative positions of the two molecules.. We show that both the function value and gradient vector can be computed analytically and apply the more advanced BFGS method in addition to the steepest descent algorithm. The algorithms are applied to compute the similarities among the 20 amino acids and biomolecules like proteins. Our computational results show that our algorithm can achieve more accuracy than previous methods and has a 6-fold speedup over the steepest descent method.     

非线性整数规划的一个新的无参数填充函数算法

离散填充函数是一种用于求解多极值优化问题最优解的一种行之有效的方法.已被证明对于求解大规模离散优化问题是有效的.本文基于改进的离散填充函数定义,构造了一个新的无参数填充函数,并在理论上给出了证明,提出了一个新的填充函数算法.该填充函数无需调节参数,而且只需极小化一次目标函数.数值结果表明,该算法是高效的、可行的.     

A Two-Term PRP-Based Descent Method

In this paper, by the use of the project of the PRP (Polak–Ribiére–Polyak) conjugate gradient direction, we develop a PRP-based descent method for solving unconstrained optimization problem. The method provides a sufficient descent direction for the objective function. Moreover, if exact line search is used, the method reduces to the standard PRP method. Under suitable conditions, we show that the method with some backtracking line search or the generalized Wolfe-type line search is globally convergent. We also report some numerical results and compare the performance of the method with some existing conjugate gradient methods. The results show that the proposed method is efficient.     

Using modifications to Grover’s Search algorithm for quantum global optimization

We study the problem of finding a global optimal solution to discrete optimization problems using a heuristic based on quantum computing methods. (Knowledge of quantum computing ideas is not necessary to read this paper.) We focus on a successful quantum computing method introduced by Baritompa, Bulger, and Wood, that we refer to as the BBW algorithm, and develop two modifications. First, we modify the BBW algorithm to achieve a dramatic speedup that lets us extend the known BBW static schedule from 33 to 43 points, thereby increasing its applicability. We further modify it by converting it from a static method to a dynamic one. Experimental results show the value of this latter modification.     

Inclusion functions and global optimization II

This paper discusses algorithms of Moore, Skelboe, Ichida, Fujii and Hansen for solving the global unconstrained optimization problem. These algorithms have been tried on computers, but a thorough theoretical discussion of their convergence properties has been missing. The discussion was started in part I of this paper (Mathematical Programming 33 (1985) 300–317) where the convergence to the global minimum was studied. The present paper is concerned with the different behaviours of these algorithms when they are used for the determination of global minimum points. The solution sets of the algorithms can be a subset of the set of global minimum points,G, a superset ofG, or exactlyG. The algorithms are applicable to a very general class of functions: functions which are continuous, and have suitable inclusion functions. The number of global minimum points can be infinite.This work was supported by the Deutsche Forschungsgemeinschaft.     

总体优化一类双参数填充函数算法的改进

求解无约束总体优化问题的一类双参数填充函数算法需要假设该问题的局部极小解的个数只有有限个，而且填充函数中参数的选取与局部极小解的谷域的半径有关．该文对其填充函数作了适当改进，使得新的填充函数算法不仅无需对问题的局部极小解的个数作假设，而且填充函数中参数的选取与局部极小解的谷域的半径无关．数值试验表明算法是有效的．     

一类不依赖于局部极小解个数的填充函数

求解无约束总体优化问题的一类单参数填充函数需要假设问题的局部极小解的个数只有有限个,而且填充函数中参数的选取与局部极小解的谷域的半径有关.本文对填充函数的定义作适当改进,而且对已有的这一类填充函数作改进,构造了一类双参数填充函数.新的填充函数不仅无须对问题的局部极小解的个数作假设,而且其中参数的选取与局部极小解的谷域的半径无关.     

Parallel Information Algorithm with Local Tuning for Solving Multidimensional GO Problems

In this paper we propose a new parallel algorithm for solving global optimization (GO) multidimensional problems. The method unifies two powerful approaches for accelerating the search: parallel computations and local tuning on the behavior of the objective function. We establish convergence conditions for the algorithm and theoretically show that the usage of local information during the global search permits to accelerate solving the problem significantly. Results of numerical experiments executed with 100 test functions are also reported.