Noisy kriging-based optimization methods: A unified implementation within the DiceOptim package

Kriging-based optimization relying on noisy evaluations of complex systems has recently motivated contributions from various research communities. Five strategies have been implemented in the DiceOptim package. The corresponding functions constitute a user-friendly tool for solving expensive noisy optimization problems in a sequential framework, while offering some flexibility for advanced users. Besides, the implementation is done in a unified environment, making this package a useful device for studying the relative performances of existing approaches depending on the experimental setup. An overview of the package structure and interface is provided, as well as a description of the strategies and some insight about the implementation challenges and the proposed solutions. The strategies are compared to some existing optimization packages on analytical test functions and show promising performances.

Optimization methods for the operating room management under uncertainty: stochastic programming vs. decomposition approach

The operating theatres are the engine of the hospitals; proper management of the operating rooms and its staff represents a great challenge for managers and its results impact directly in the budget of the hospital. This work presents a MILP model for the efficient schedule of multiple surgeries in Operating Rooms (ORs) during a working day. This model considers multiple surgeons and ORs and different types of surgeries. Stochastic strategies are also implemented for taking into account the uncertain in surgery durations (pre-incision, incision, post-incision times). In addition, a heuristic-based methods and a MILP decomposition approach is proposed for solving large-scale ORs scheduling problems in computational efficient way. All these computer-aided strategies has been implemented in AIMMS, as an advanced modeling and optimization software, developing a user friendly solution tool for the operating room management under uncertainty.

A theoretical comparison of two optimization methods for radiofrequency drive schemes in high frequency MRI resonators

In this paper, numerical simulations are used in an attempt to find optimal Source profiles for high frequency radiofrequency (RF) volume coils. Biologically loaded, shielded/unshielded circular and elliptical birdcage coils operating at 170 MHz, 300 MHz and 470 MHz are modelled using the FDTD method for both 2D and 3D cases. Taking advantage of the fact that some aspects of the electromagnetic system are linear, two approaches have been proposed for the determination of the drives for individual elements in the RF resonator. The first method is an iterative optimization technique with a kernel for the evaluation of RF fields inside an imaging plane of a human head model using pre-characterized sensitivity profiles of the individual rungs of a resonator; the second method is a regularization-based technique. In the second approach, a sensitivity matrix is explicitly constructed and a regularization procedure is employed to solve the ill-posed problem. Test simulations show that both methods can improve the B-1-field homogeneity in both focused and non-focused scenarios. While the regularization-based method is more efficient, the first optimization method is more flexible as it can take into account other issues such as controlling SAR or reshaping the resonator structures. It is hoped that these schemes and their extensions will be useful for the determination of multi-element RF drives in a variety of applications.

Purification of phosphoric acid by solvent extraction

This work follows a feasibility study (187) which suggested that a process for purifying wet-process phosphoric acid by solvent extraction should be economically viable. The work was divided into two main areas, (i) chemical and physical measurements on the three-phase system, with or without impurities; (ii) process simulation and optimization. The object was to test the process technically and economically and to optimise the type of solvent. The chemical equilibria and distribution curves for the system water - phosphoric acid - solvent for the solvents n-amyl alcohol, tri-n-butyl phosphate, di-isopropyl ether and methyl isobutyl ketone have been determined. Both pure phosphoric acid and acid containing known amounts of naturally occurring impurities (Fe P0 4 , A1P0 4 , Ca3(P04)Z and Mg 3(P0 4 )Z) were examined. The hydrodynamic characteristics of the systems were also studied. The experimental results obtained for drop size distribution were compared with those obtainable from Hinze's equation (32) and it was found that they deviated by an amount related to the turbulence. A comprehensive literature survey on the purification of wet-process phosphoric acid by organic solvents has been made. The literature regarding solvent extraction fundamentals and equipment and optimization methods for the envisaged process was also reviewed. A modified form of the Kremser-Brown and Souders equation to calculate the number of contact stages was derived. The modification takes into account the special nature of phosphoric acid distribution curves in the studied systems. The process flow-sheet was developed and simulated. Powell's direct search optimization method was selected in conjunction with the linear search algorithm of Davies, Swann and Campey. The objective function was defined as the total annual manufacturing cost and the program was employed to find the optimum operating conditions for anyone of the chosen solvents. The final results demonstrated the following order of feasibility to purify wet-process acid: di-isopropyl ether, methylisobutyl ketone, n-amyl alcohol and tri-n-butyl phosphate.

Direct multisearch for multiobjective optimization

In practical applications of optimization it is common to have several conflicting objective functions to optimize. Frequently, these functions are subject to noise or can be of black-box type, preventing the use of derivative-based techniques. We propose a novel multiobjective derivative-free methodology, calling it direct multisearch (DMS), which does not aggregate any of the objective functions. Our framework is inspired by the search/poll paradigm of direct-search methods of directional type and uses the concept of Pareto dominance to maintain a list of nondominated points (from which the new iterates or poll centers are chosen). The aim of our method is to generate as many points in the Pareto front as possible from the polling procedure itself, while keeping the whole framework general enough to accommodate other disseminating strategies, in particular, when using the (here also) optional search step. DMS generalizes to multiobjective optimization (MOO) all direct-search methods of directional type. We prove under the common assumptions used in direct search for single objective optimization that at least one limit point of the sequence of iterates generated by DMS lies in (a stationary form of) the Pareto front. However, extensive computational experience has shown that our methodology has an impressive capability of generating the whole Pareto front, even without using a search step. Two by-products of this paper are (i) the development of a collection of test problems for MOO and (ii) the extension of performance and data profiles to MOO, allowing a comparison of several solvers on a large set of test problems, in terms of their efficiency and robustness to determine Pareto fronts.

Derivative-free optimization and filter methods to solve nonlinear constrained problems

In real optimization problems, usually the analytical expression of the objective function is not known, nor its derivatives, or they are complex. In these cases it becomes essential to use optimization methods where the calculation of the derivatives, or the verification of their existence, is not necessary: the Direct Search Methods or Derivative-free Methods are one solution. When the problem has constraints, penalty functions are often used. Unfortunately the choice of the penalty parameters is, frequently, very difficult, because most strategies for choosing it are heuristics strategies. As an alternative to penalty function appeared the filter methods. A filter algorithm introduces a function that aggregates the constrained violations and constructs a biobjective problem. In this problem the step is accepted if it either reduces the objective function or the constrained violation. This implies that the filter methods are less parameter dependent than a penalty function. In this work, we present a new direct search method, based on simplex methods, for general constrained optimization that combines the features of the simplex method and filter methods. This method does not compute or approximate any derivatives, penalty constants or Lagrange multipliers. The basic idea of simplex filter algorithm is to construct an initial simplex and use the simplex to drive the search. We illustrate the behavior of our algorithm through some examples. The proposed methods were implemented in Java.

Node adaptation for global collocation with radial basis functions using direct multisearch for multiobjective optimization

Meshless methods are used for their capability of producing excellent solutions without requiring a mesh, avoiding mesh related problems encountered in other numerical methods, such as finite elements. However, node placement is still an open question, specially in strong form collocation meshless methods. The number of used nodes can have a big influence on matrix size and therefore produce ill-conditioned matrices. In order to optimize node position and number, a direct multisearch technique for multiobjective optimization is used to optimize node distribution in the global collocation method using radial basis functions. The optimization method is applied to the bending of isotropic simply supported plates. Using as a starting condition a uniformly distributed grid, results show that the method is capable of reducing the number of nodes in the grid without compromising the accuracy of the solution. (C) 2013 Elsevier Ltd. All rights reserved.

Derivative-free nonlinear optimization filter simplex

The filter method is a technique for solving nonlinear programming problems. The filter algorithm has two phases in each iteration. The first one reduces a measure of infeasibility, while in the second the objective function value is reduced. In real optimization problems, usually the objective function is not differentiable or its derivatives are unknown. In these cases it becomes essential to use optimization methods where the calculation of the derivatives or the verification of their existence is not necessary: direct search methods or derivative-free methods are examples of such techniques. In this work we present a new direct search method, based on simplex methods, for general constrained optimization that combines the features of simplex and filter methods. This method neither computes nor approximates derivatives, penalty constants or Lagrange multipliers.

Classification of some penalty methods

Optimization problems arise in science, engineering, economy, etc. and we need to find the best solutions for each reality. The methods used to solve these problems depend on several factors, including the amount and type of accessible information, the available algorithms for solving them, and, obviously, the intrinsic characteristics of the problem. There are many kinds of optimization problems and, consequently, many kinds of methods to solve them. When the involved functions are nonlinear and their derivatives are not known or are very difficult to calculate, these methods are more rare. These kinds of functions are frequently called black box functions. To solve such problems without constraints (unconstrained optimization), we can use direct search methods. These methods do not require any derivatives or approximations of them. But when the problem has constraints (nonlinear programming problems) and, additionally, the constraint functions are black box functions, it is much more difficult to find the most appropriate method. Penalty methods can then be used. They transform the original problem into a sequence of other problems, derived from the initial, all without constraints. Then this sequence of problems (without constraints) can be solved using the methods available for unconstrained optimization. In this chapter, we present a classification of some of the existing penalty methods and describe some of their assumptions and limitations. These methods allow the solving of optimization problems with continuous, discrete, and mixing constraints, without requiring continuity, differentiability, or convexity. Thus, penalty methods can be used as the first step in the resolution of constrained problems, by means of methods that typically are used by unconstrained problems. We also discuss a new class of penalty methods for nonlinear optimization, which adjust the penalty parameter dynamically.

Improvement of a filters method in a derivative free optimization

Constraints nonlinear optimization problems can be solved using penalty or barrier functions. This strategy, based on solving the problems without constraints obtained from the original problem, have shown to be e ective, particularly when used with direct search methods. An alternative to solve the previous problems is the lters method. The lters method introduced by Fletcher and Ley er in 2002, , has been widely used to solve problems of the type mentioned above. These methods use a strategy di erent from the barrier or penalty functions. The previous functions de ne a new one that combine the objective function and the constraints, while the lters method treat optimization problems as a bi-objective problems that minimize the objective function and a function that aggregates the constraints. Motivated by the work of Audet and Dennis in 2004, using lters method with derivative-free algorithms, the authors developed works where other direct search meth- ods were used, combining their potential with the lters method. More recently. In a new variant of these methods was presented, where it some alternative aggregation restrictions for the construction of lters were proposed. This paper presents a variant of the lters method, more robust than the previous ones, that has been implemented with a safeguard procedure where values of the function and constraints are interlinked and not treated completely independently.

Multiobjective optimization for node adaptation in the analysis of composite plates using a meshless collocation method

The bending of simply supported composite plates is analyzed using a direct collocation meshless numerical method. In order to optimize node distribution the Direct MultiSearch (DMS) for multi-objective optimization method is applied. In addition, the method optimizes the shape parameter in radial basis functions. The optimization algorithm was able to find good solutions for a large variety of nodes distribution.

Robust parameter settings of evolutionary algorithms for the optimisation of agricultural systems models

Numerical optimisation methods are being more commonly applied to agricultural systems models, to identify the most profitable management strategies. The available optimisation algorithms are reviewed and compared, with literature and our studies identifying evolutionary algorithms (including genetic algorithms) as superior in this regard to simulated annealing, tabu search, hill-climbing, and direct-search methods. Results of a complex beef property optimisation, using a real-value genetic algorithm, are presented. The relative contributions of the range of operational options and parameters of this method are discussed, and general recommendations listed to assist practitioners applying evolutionary algorithms to the solution of agricultural systems. (C) 2001 Elsevier Science Ltd. All rights reserved.

Optimization techniques for power distribution planning with uncertainties: a comparative study

Important research effort has been devoted to the topic of optimal planning of distribution systems. The non linear nature of the system, the need to consider a large number of scenarios and the increasing necessity to deal with uncertainties make optimal planning in distribution systems a difficult task. Heuristic techniques approaches have been proposed to deal with these issues, overcoming some of the inherent difficulties of classic methodologies. This paper considers several methodologies used to address planning problems of electrical power distribution networks, namely mixedinteger linear programming (MILP), ant colony algorithms (AC), genetic algorithms (GA), tabu search (TS), branch exchange (BE), simulated annealing (SA) and the Bender´s decomposition deterministic non-linear optimization technique (BD). Adequacy of theses techniques to deal with uncertainties is discussed. The behaviour of each optimization technique is compared from the point of view of the obtained solution and of the methodology performance. The paper presents results of the application of these optimization techniques to a real case of a 10-kV electrical distribution system with 201 nodes that feeds an urban area.

Adaptive Penalty and Barrier function based on Fuzzy Logic

Optimization methods have been used in many areas of knowledge, such as Engineering, Statistics, Chemistry, among others, to solve optimization problems. In many cases it is not possible to use derivative methods, due to the characteristics of the problem to be solved and/or its constraints, for example if the involved functions are non-smooth and/or their derivatives are not know. To solve this type of problems a Java based API has been implemented, which includes only derivative-free optimization methods, and that can be used to solve both constrained and unconstrained problems. For solving constrained problems, the classic Penalty and Barrier functions were included in the API. In this paper a new approach to Penalty and Barrier functions, based on Fuzzy Logic, is proposed. Two penalty functions, that impose a progressive penalization to solutions that violate the constraints, are discussed. The implemented functions impose a low penalization when the violation of the constraints is low and a heavy penalty when the violation is high. Numerical results, obtained using twenty-eight test problems, comparing the proposed Fuzzy Logic based functions to six of the classic Penalty and Barrier functions are presented. Considering the achieved results, it can be concluded that the proposed penalty functions besides being very robust also have a very good performance.