Class Of Global Optimization Problems Research Articles

Convexification for a Class of Global Optimization Problems with C$$^{1,1}$$ Functions

Convexification for a Class of Global Optimization Problems with C$$^{1,1}$$ Functions

On Convexification for a Class of Global Optimization Problems

On Convexification for a Class of Global Optimization Problems

Multi-level evolution strategies for high-resolution black-box control

This paper introduces a multi-level (m-lev) mechanism into Evolution Strategies (ESs) in order to address a class of global optimization problems that could benefit from fine discretization of their decision variables. Such problems arise in engineering and scientific applications, which possess a multi-resolution control nature, and thus may be formulated either by means of low-resolution variants (providing coarser approximations with presumably lower accuracy for the general problem) or by high-resolution controls. A particular scientific application concerns practical Quantum Control (QC) problems, whose targeted optimal controls may be discretized to increasingly higher resolution, which in turn carries the potential to obtain better control yields. However, state-of-the-art derivative-free optimization heuristics for high-resolution formulations nominally call for an impractically large number of objective function calls. Therefore, an effective algorithmic treatment for such problems is needed. We introduce a framework with an automated scheme to facilitate guided-search over increasingly finer levels of control resolution for the optimization problem, whose on-the-fly learned parameters require careful adaptation. We instantiate the proposed m-lev self-adaptive ES framework by two specific strategies, namely the classical elitist single-child (1+1)-ES and the non-elitist multi-child derandomized $$(\mu _W,\lambda )$$ -sep-CMA-ES. We first show that the approach is suitable by simulation-based optimization of QC systems which were heretofore viewed as too complex to address. We also present a laboratory proof-of-concept for the proposed approach on a basic experimental QC system objective.

A deterministic method for continuous global optimization using a dense curve

In this paper, we develop a new approach for solving a large class of global optimization problems for objective functions which are only continuous on a rectangle of Rn. This method is based on the reducing transformation technique by running in the feasible domain a single parametrized Lissajous curve, which becomes increasingly denser and progressively fills the feasible domain. By means of the one-dimensional Evtushenko algorithm, we realize a mixed method which explores the feasible domain. To speed up the mixed exploration algorithm, we have incorporated a DIRECT local search type algorithm to explore promising regions. This method converges in a finite number of iterations to the global minimum within a prescribed accuracy ε>0. Simulations on some typical test problems with diverse properties and different dimensions indicate that the algorithm is promising and competitive.

A vector linear programming approach for certain global optimization problems

Global optimization problems with a quasi-concave objective function and linear constraints are studied. We point out that various other classes of global optimization problems can be expressed in this way. We present two algorithms, which can be seen as slight modifications of Benson-type algorithms for multiple objective linear programs (MOLP). The modification of the MOLP algorithms results in a more efficient treatment of the studied optimization problems. This paper generalizes results of Schulz and Mittal on quasi-concave problems and Shao and Ehrgott on multiplicative linear programs. Furthermore, it improves results of L\"ohne and Wagner on minimizing the difference $f=g-h$ of two convex functions $g$, $h$ where either $g$ or $h$ is polyhedral. Numerical examples are given and the results are compared with the global optimization software BARON.

On strong homogeneity of a class of global optimization algorithms working with infinite and infinitesimal scales

The necessity to find the global optimum of multiextremal functions arises in many applied problems where finding local solutions is insufficient. One of the desirable properties of global optimization methods is strong homogeneity meaning that a method produces the same sequences of points where the objective function is evaluated independently both of multiplication of the function by a scaling constant and of adding a shifting constant. In this paper, several aspects of global optimization using strongly homogeneous methods are considered. First, it is shown that even if a method possesses this property theoretically, numerically very small and large scaling constants can lead to ill-conditioning of the scaled problem. Second, a new class of global optimization problems where the objective function can have not only finite but also infinite or infinitesimal Lipschitz constants is introduced. Third, the strong homogeneity of several Lipschitz global optimization algorithms is studied in the framework of the Infinity Computing paradigm allowing one to work numerically with a variety of infinities and infinitesimals. Fourth, it is proved that a class of efficient univariate methods enjoys this property for finite, infinite and infinitesimal scaling and shifting constants. Finally, it is shown that in certain cases the usage of numerical infinities and infinitesimals can avoid ill-conditioning produced by scaling. Numerical experiments illustrating theoretical results are described.

On solving the sum-of-ratios problem

This paper addresses the development of efficient global search methods for fractional programming problems. Such problems are, in general, nonconvex (with numerous local extremums) and belong to a class of global optimization problems. First, we reduce a rather general fractional programming problem with d.c. functions to solving an equation with a vector parameter that satisfies some nonnegativity assumption. This theorem allows the justified use of the generalized Dinkelbach’s approach for solving fractional programming problems with a d.c. goal function. Based on solving of some d.c. minimization problem, we developed a global search algorithm for fractional programming problems, which was tested on a set of low-dimensional test problems taken from the literature as well as on randomly generated problems with up to 200 variables or 200 terms in the sum.

A parallel metaheuristic for large mixed-integer dynamic optimization problems, with applications in computational biology.

BackgroundWe consider a general class of global optimization problems dealing with nonlinear dynamic models. Although this class is relevant to many areas of science and engineering, here we are interested in applying this framework to the reverse engineering problem in computational systems biology, which yields very large mixed-integer dynamic optimization (MIDO) problems. In particular, we consider the framework of logic-based ordinary differential equations (ODEs).MethodsWe present saCeSS2, a parallel method for the solution of this class of problems. This method is based on an parallel cooperative scatter search metaheuristic, with new mechanisms of self-adaptation and specific extensions to handle large mixed-integer problems. We have paid special attention to the avoidance of convergence stagnation using adaptive cooperation strategies tailored to this class of problems.ResultsWe illustrate its performance with a set of three very challenging case studies from the domain of dynamic modelling of cell signaling. The simpler case study considers a synthetic signaling pathway and has 84 continuous and 34 binary decision variables. A second case study considers the dynamic modeling of signaling in liver cancer using high-throughput data, and has 135 continuous and 109 binaries decision variables. The third case study is an extremely difficult problem related with breast cancer, involving 690 continuous and 138 binary decision variables. We report computational results obtained in different infrastructures, including a local cluster, a large supercomputer and a public cloud platform. Interestingly, the results show how the cooperation of individual parallel searches modifies the systemic properties of the sequential algorithm, achieving superlinear speedups compared to an individual search (e.g. speedups of 15 with 10 cores), and significantly improving (above a 60%) the performance with respect to a non-cooperative parallel scheme. The scalability of the method is also good (tests were performed using up to 300 cores).ConclusionsThese results demonstrate that saCeSS2 can be used to successfully reverse engineer large dynamic models of complex biological pathways. Further, these results open up new possibilities for other MIDO-based large-scale applications in the life sciences such as metabolic engineering, synthetic biology, drug scheduling.

Continuous global optimization through the generation of parametric curves

In this paper we develop a new approach for solving a large class of global optimization problems. The objective function is only continuous, non-smooth and non-Lipschitzian, defined on a rectangle of Rn. This approach is based on the generation, in the feasible set, of a family of parametrized curves satisfying certain properties combined with the one-dimensional Evtushenko algorithm. To accelerate the corresponding mixed algorithm, we have incorporated in a variant a Pattern Search-type deterministic local optimization method and in another variant a new stochastic local optimization method. Both variants have been applied to several typical test problems. A comparison with some well known methods is highlighted through numerical experiments.

Energy Efficient Motion Control for a Light Rail Vehicle Using The Big Bang Big Crunch Algorithm

Abstract: In this manuscript, an emerging evoluitonary optimization technique, Big Bang - Big Crunch algorithm, is used for an energy-efficient train operation in light rail network of Eskisehir is studied. Cruising and coasting, two basic motion phases of train, should be taken into consideration in order to decrease energy consumption. Determining the optimal switching points from one motion phase into another is key in the energy saving. Switching points for optimal train operation are modeled for the Big Bang Big Crunch algorithm platform and globally optimum set of switching points is obtained as its output, which has proved to be suitable for solving a wide class of global optimization problems. Obtained results are verified through simulations of the operation for five consecutive stations between Buyukdere and Stadyum. Test paths are created considering actual grade and track length. Also certain real life constraints are taken into account such as punctuality and maximum speed limit. It is shown that using the optimal switching points in the simulations results in a significant energy saving.

Range division and contraction algorithm for a class of global optimization problems

This article presents a range division and contraction algorithm for a class of global optimization problems. In the algorithm, the original problem is first converted into an equivalent monotonic optimization problem whose objective function is just a simple univariate. By exploiting the particularity of this monotonicity, the variable bound in a particular node is then tightened and the bounds on the constraints are calculated to remove the region which doesn’t contain optimal solutions. The proposed method can reach an approximate solution within an acceptable error, in which such solution is adequately guaranteed to be feasible and to be close to the actual global optimal solution. Several numerical examples are given to illustrate the feasibility and efficiency of the present algorithm.

Billiard walk - a new sampling algorithm for control and optimization

Hit-and-Run is known to be one of the best versions of Markov Chain Monte Carlo sampler. Nevertheless, in practice the number of iterations required to achieve uniformly distributed samples is rather high. We propose new random walk algorithm based on billiard trajectories and prove its asymptotic uniformity. Numerical experiments demonstrate much faster convergence to uniform distribution for Billiard Walk algorithm compared to Hit-and- Run. We discuss a class of global optimization problems that can be efficiently solved with Monte Carlo sampler.

Problems with resource allocation constraints and optimization over the efficient set

The paper studies a nonlinear optimization problem under resource allocation constraints. Using quasi-gradient duality it is shown that the feasible set of the problem is a singleton (in the case of a single resource) or the set of Pareto efficient solutions of an associated vector maximization problem (in the case of $$k>1$$ resources). As a result, a nonlinear optimization problem under resource allocation constraints reduces to an optimization over the efficient set. The latter problem can further be converted into a quasiconvex maximization over a compact convex subset of $$\mathbb{R }^k_+.$$ Alternatively, it can be approached as a bilevel program and converted into a monotonic optimization problem in $$\mathbb{R }^k_+.$$ In either approach the converted problem falls into a common class of global optimization problems for which several practical solution methods exist when the number $$k$$ of resources is relatively small, as it often occurs.

Solution to Singular Optimal Control by Backward Differential Flow

This paper presents a backward differential flow for solving singular optimal control problems. By using Krotov equivalent transformation, the cost functional is converted to a class of global optimization problems. Some properties of the flow are given to reveal the significant relationship between the dynamic of the flow and the geometry of the feasible set. The proposed method is also used in solving a class of variational problems. Some examples are illustrated.

Approximation algorithm for a class of global optimization problems

In this paper we develop and derive the computational cost of an $${\varepsilon}$$ -approximation algorithm for a class of global optimization problems, where a suitably defined composition of some ratio functions is minimized over a convex set. The result extends a previous one about a class of Linear Fractional/Multiplicative problems.

A new hybrid method for solving global optimization problem

In this paper we present a new hybrid method, called the SASP method. The purpose of this method is the hybridization of the simulated annealing (SA) with the descent method, where we estimate the gradient using simultaneous perturbation. Firstly, the new hybrid method finds a local minimum using the descent method, then SA is executed in order to escape from the currently discovered local minimum to a better one, from which the descent method restarts a new local search, and so on until convergence. The new hybrid method can be widely applied to a class of global optimization problems for continuous functions with constraints. Experiments on 30 benchmark functions, including high dimensional functions, show that the new method is able to find near optimal solutions efficiently. In addition, its performance as a viable optimization method is demonstrated by comparing it with other existing algorithms. Numerical results improve the robustness and efficiency of the method presented.

Global Descent Method for Global Optimization

This paper develops a novel method—the global descent method—for solving a general class of global optimization problems. This method moves from one local minimizer of the objective function $f$ to a better one at each iteration with the help of an auxiliary function termed the global descent function. The global descent function is not only guaranteed to have a local minimizer $x'$ over the problem domain in $\mathbb{R}^n$ but also ensures that each of its local minimizers is located in some neighborhoods of a better minimizer of $f$ with $f(x')<f(x^*)$. These features of the global descent function enable a global descent to be achieved at each iteration using only local descent methods. Computational experiments conducted on several test problems with up to 1000 variables demonstrate the applicability of the proposed method. Furthermore, numerical comparison experiments carried out with GAMS/BARON on several test problems also justify the efficiency and effectiveness of the proposed method.

A sufficient and necessary condition for global optimization

We consider a class of global optimization problems, in which the objective function is Lebesgue integrable. In this paper, we present a sufficient and necessary condition for computing the essential infimum. This optimality condition can be used to design algorithms for solving the global optimization problem.

Optimal solutions to a class of nonconvex minimization problems with linear inequality constraints

This paper presents a canonical duality theory and optimal solutions to a class of global optimization problems subjected to linear inequality constraints. By using the canonical dual transformation developed recently, a canonical dual problem is formulated, which is perfectly dual to the primal problem. The global minmizer can be identified by the triality theory. Results show that if the global extrema of the original problem are located on the boundary of the primal feasible space, the dual solution should be interior point of the dual feasible set. Several examples are illustrated to show how this theory works.

A hybrid genetic algorithm for a class of global optimization problems with box constraints

In this paper, a new hybrid genetic algorithm is proposed, which combines the genetic algorithm with hill-climbing search steps differently from some former algorithms. The new algorithm can be widely applied to a class of global optimization problems for continuous functions with box constraints. Finally, numerical examples show that this algorithm can yield the global optimum with high efficiency.