Infeasible Regions Research Articles

A new deterministic global optimization method for general twice-differentiable constrained nonlinear programming problems

A deterministic global optimization method that is applicable to general nonlinear programming problems composed of twice-differentiable objective and constraint functions is proposed. The method hybridizes the branch-and-bound algorithm and a convex cut function (CCF). For a given subregion, the difference of a convex underestimator that does not need an iterative local optimizer to determine the lower bound of the objective function is generated. If the obtained lower bound is located in an infeasible region, then the CCF is generated for constraints to cut this region. The cutting region generated by the CCF forms a hyperellipsoid and serves as the basis of a discarding rule for the selected subregion. However, the convergence rate decreases as the number of cutting regions increases. To accelerate the convergence rate, an inclusion relation between two hyperellipsoids should be applied in order to reduce the number of cutting regions. It is shown that the two-hyperellipsoid inclusion relation is determined by maximizing a quadratic function over a sphere, which is a special case of a trust region subproblem. The proposed method is applied to twelve nonlinear programming test problems and five engineering design problems. Numerical results show that the proposed method converges in a finite calculation time and produces accurate solutions.

A new global optimization method for univariate constrained twice-differentiable NLP problems

In this paper, a new global optimization method is proposed for an optimization problem with twice-differentiable objective and constraint functions of a single variable. The method employs a difference of convex underestimator and a convex cut function, where the former is a continuous piecewise concave quadratic function, and the latter is a convex quadratic function. The main objectives of this research are to determine a quadratic concave underestimator that does not need an iterative local optimizer to determine the lower bounding value of the objective function and to determine a convex cut function that effectively detects infeasible regions for nonconvex constraints. The proposed method is proven to have a finite ?-convergence to locate the global optimum point. The numerical experiments indicate that the proposed method competes with another covering method, the index branch-and-bound algorithm, which uses the Lipschitz constant.

Penalty guided genetic search for redundancy optimization in multi-state series-parallel power system

This paper presents a genetic algorithm (GA) for parallel redundancy optimization in series-parallel power systems exhibiting multi-state behavior, optimizing the reliability subject to constraints. The components are binary and chosen from a list of products available in the market, and are being characterized by their feeding capacity, reliability, cost and weight. System reliability is defined as the ability to satisfy consumer demand and is presented as a piecewise cumulative load curve. In GA, to handle infeasible solutions penalty strategies are used. Penalty technique keep a certain amount of infeasible solutions in each generation so as to enforce genetic search towards an optimal solution from sides of, both, feasible and infeasible regions. We here present a dynamic adaptive penalty function which helps the algorithm to search efficiently for optimal/near optimal solution. To evaluate system reliability, a fast procedure, based on universal generating function, is used. An example considering a multi-state series-parallel power system is solved considering both homogeneous and heterogeneous types of redundancy. Also an example considering price discounts is solved. The effectiveness of the penalty function and the proposed algorithm is studied and shown graphically.

Infinite/infinite analysis as a tool for an early oriented synthesis of a reactive pressure swing distillation

The study contributes to the characterization of an original reactive pressure swing distillation system. The methyl acetate (MeAc) transesterification with ethanol (EtOH) to produce methanol (MeOH) and ethyl acetate (EtAc) is shown as illustrative example. The streams outside the units are evaluated by the ∞/∞ analysis to provide insights on the process behavior. Two simpler systems with recycling stream are also presented. The ∞/∞ analysis allows checking the interrelation of the system streams without any column design consideration. Unfeasible regions, low limit values, multiplicity regions, discontinuities, control difficulties, recommendable operation conditions and column profile combinations are predicted and discussed. All these information are useful to establish an early and suitable system design strategy.

IAs based approach for reliability redundancy allocation problems

Nonlinearly mixed-integer reliability design problems are investigated in this paper where both the number of redundancy components and the corresponding component reliability in each subsystem are to be decided simultaneously so as to maximize the reliability of system. The reliability design problems have been studied in the literature for decades, usually using mathematical programming or heuristic/metaheuristic optimization approaches. The difficulties confronted for both methodologies are to maintain feasibility with respect to three nonlinear constraints, namely, cost, weight and volume related constraints. A penalty guided artificial immune algorithm is presented for solving such mixed-integer reliability design problems. It can search over promising feasible and infeasible regions to find the feasible optimal/near optimal solution effectively and efficiently. Numerical examples indicate that the proposed approach performs well for the reliability-redundant allocation design problems considered in this paper. As reported, solutions obtained by the proposed approach are as well as or better than the previously best-known solutions.

Automatic loop shaping in QFT using hybrid optimization and constraint propagation techniques

AbstractAn efficient method for automatic loop shaping in Quantitative Feedback Theory (QFT) is proposed. The proposed method uses hybrid optimization and constraint propagation techniques. The hybrid optimization part combines interval global optimization and nonlinear local optimization methods. To supplement the optimization tools, a quick solution approach is developed based on ideas of constraint propagation. The quick solution is found to be quite effective in discarding infeasible controller parameter regions in the optimization search. The proposed overall method is demonstrated on an example taken from the QFT literature where it automatically synthesizes a QFT controller of given structure in quick time. Copyright © 2006 John Wiley & Sons, Ltd.

A refined algorithm for multisensor image registration based on pixel migration

Multimodality image registration via pixel migration is a powerful approach. However, it suffers from a serious problem--the global maximum on the sum of squared gradient magnitude (SSG) surface does not correspond to the correct solution of registration. To solve the problem, we partition the search space into feasible and infeasible regions. The genetic algorithm (global optimizer) is used to obtain a good initial estimate of registration parameters and followed by a fast refining with Powell's approach (local optimizer). The experimental results demonstrate that the use of this modified pixel migration algorithm on multisensor image registration is very effective.

A 3-flip neighborhood local search for the set covering problem

The set covering problem (SCP) calls for a minimum cost family of subsets from n given subsets, which together covers the entire ground set. In this paper, we propose a local search algorithm for SCP, which has the following three characteristics. (1) The use of 3-flip neighborhood, which is the set of solutions obtainable from the current solution by exchanging at most three subsets. As the size of 3-flip neighborhood is O( n 3), the neighborhood search becomes expensive if implemented naively. To overcome this, we propose an efficient implementation that reduces the number of candidates in the neighborhood without sacrificing the solution quality. (2) We allow the search to visit the infeasible region, and incorporate the strategic oscillation technique realized by adaptive control of penalty weights. (3) The size reduction of the problem by using the information from the Lagrangian relaxation is incorporated, which is indispensable for solving very large instances. According to computational comparisons on benchmark instances with other existing heuristic algorithms for SCP, our algorithm performs quite effectively for various types of problems, especially for very large-scale instances.

Genetic algorithm with adaptive and dynamic penalty functions for the selection of cleaner production measures: a constrained optimization problem

This paper presents a new approach of genetic algorithm (GA) to solve the constrained optimization problem. In a constrained optimization problem, feasible and infeasible regions occupy the search space. The infeasible regions consist of the solutions that violate the constraint. Oftentimes classical genetic operators generate infeasible or invalid chromosomes. This situation takes a turn for the worse when infeasible chromosomes alone occupy the whole population. To address this problem, dynamic and adaptive penalty functions are proposed for the GA search process. This is a novel strategy because it will attempt to transform the constrained problem into an unconstrained problem by penalizing the GA fitness function dynamically and adaptively. New equations describing these functions are presented and tested. The effects of the proposed functions developed have been investigated and tested using different GA parameters such as mutation and crossover. Comparisons of the performance of the proposed adaptive and dynamic penalty functions with traditional static penalty functions are presented. The result from the experiments show that the proposed functions developed are more accurate, efficient, robust and easy to implement. The algorithms developed in this research can be applied to evaluate environmental impacts from process operations.

A simulated annealing approach to the traveling tournament problem

Automating the scheduling of sport leagues has received considerable attention in recent years, as these applications involve significant revenues and generate challenging combinatorial optimization problems. This paper considers the traveling tournament problem (TTP) which abstracts the salient features of major league baseball (MLB) in the United States. It proposes a simulated annealing algorithm (TTSA) for the TTP that explores both feasible and infeasible schedules, uses a large neighborhood with complex moves, and includes advanced techniques such as strategic oscillation and reheats to balance the exploration of the feasible and infeasible regions and to escape local minima at very low temperatures. TTSA matches the best-known solutions on the small instances of the TTP and produces significant improvements over previous approaches on the larger instances. Moreover, TTSA is shown to be robust, because its worst solution quality over 50 runs is always smaller or equal to the best-known solutions.

Recursive pointwise design for nonlinear systems

This paper presents a recursive pointwise design (RPD) method for a class of nonlinear systems represented by x(t)=f(x(t))+g(x(t))u(t). A main feature of the RPD method is to recursively design a stable controller by using pointwise information of a system. The design philosophy is that f(x(t)) and g(x(t)) can be approximated as constant vectors in very small local state spaces. Based on the design philosophy, we numerically determine constant control inputs in very small local state spaces by solving linear matrix inequalities (LMIs) derived in this paper. The designed controller switches to another constant control input when the states move to another local state space. Although the design philosophy is simple and natural, the controller does not always guarantee the stability of the original nonlinear system x(t)=f(x(t))+g(x(t))u(t). Therefore, this paper gives ideas of compensating the approximation caused by the design philosophy. After addressing outline of the pointwise design, we provide design conditions that exactly guarantee the stability of the original system. The controller design conditions require to give the maximum and minimum values of elements in the functions f(x(t)) and g(x(t)) in each local state - space. Therefore, we also present design conditions for unknown cases of the maximum and minimum values. Furthermore, we construct an effective design procedure using the pointwise design. A feature of the design procedure is to subdivide only infeasible regions and to solve LMIs again only for the subdivided infeasible regions. The recursive procedure saves effort to design a controller. A design example demonstrates the utility of the RPD method

An intelligent moving object optimization algorithm for design problems with mixed variables, mixed constraints and multiple objectives

This paper presents an optimization algorithm for engineering design problems having a mix of continuous, discrete and integer variables; a mix of linear, non-linear, differentiable, non-differential, equality, inequality and even discontinuous design constraints; and conflicting multiple design objectives. The intelligent movement of objects (vertices and compounds) is simulated in the algorithm based on a Nelder–Mead simplex with added features to handle variable types, bound and design constraints, local optima, search initiation from an infeasible region and numerical instability, which are the common requirements for large-scale, complex optimization problems in various engineering and business disciplines. The algorithm is called an INTElligent Moving Object algorithm and tested for a wide range of benchmark problems. Validation results for several examples, which are manageable within the scope of this paper, are presented herein. Satisfactory results have been obtained for all the test problems, hence, highlighting the benefits of the proposed method.

Penalty Function Approach in Heuristic Algorithms for Constrained Redundancy Reliability Optimization

To solve the problem of constrained redundancy reliability optimization, several heuristic algorithms have been developed in the literature. Most of these algorithms search for the solutions remaining within the feasible boundary e.g. [15], [20]. Perhaps the most interesting & efficient heuristic algorithm in terms of solution quality is that given by KYA, in which the search is made not only in the feasible region but also into the bounded infeasible region by making an excursion, which returns to the feasible region with a possibly improved solution. In this paper, a heuristic algorithm based on the penalty function approach is proposed to solve the constrained redundancy optimization problem for complex systems. An excursion is made into the infeasible region, but an adaptive penalty function helps the search not to go too far into the infeasible region. Thus, promising feasible & infeasible regions of the search space are explored efficiently & effectively to identify finally an optimal or near optimal solution. Computational experiments are conducted on 11 sets of problems (10 with linear constraints, and 1 with nonlinear constraints); each with 10 different randomly generated initial solutions. Comparison is made between the proposed algorithm P-Alg, N-N algorithm [15], Shi algorithm [20], and KYA [9] . It is observed that P-Alg performs consistently better than others, showing an overall improvement in various measures of performance. Besides, as P-Alg does not require any assumptions on the nature of the objective & constraint functions, it can solve a wide variety of problems.

노이즈 필터링을 적용한 반응표면 기반 순차적 근사 최적화

In this paper, a new method for constrained optimization of noisy functions is proposed. In approximate optimization using response surface methods, if constraints have severe noise, the approximate feasible region defined by approximate constraints is apt to include some of the infeasible region defined by actual constraints. This can cause the approximate optimum to converge into the infeasible region. In the proposed method, the approximate optimization is performed with the approximate constraints shifted by their deviations, which are calculated using a diagonal quadratic response surface method. This can prevent the approximate optimum from converging into the infeasible region. To fit the objective and constraints into diagonal quadratic models, we select the center and 4 additional points along each axis of design variables as experimental points. The deviation of each function is calculated using the differences between the real and approximate function values at the experimental points. A sequential approximate optimization technique based on the trust region algorithm is adopted to manage approximate models. The proposed approach is validated by solving some design problems. The results of the problems show the effectiveness of the proposed method.

A Local-Search-Based Heuristic for the Demand-Constrained Multidimensional Knapsack Problem

We consider an extension of the 0–1 multidimensional knapsack problem in which there are greater-than-or-equal-to inequalities, called demand constraints, in addition to the standard less-than-or-equal-to constraints. Moreover, the objective function coefficients are not constrained in sign. This problem is worth considering because it is embedded in models of practical application, it has an intriguing combinatorial structure, and it appears to be a challenging problem for commercial ILP solvers. Our approach is based on a nested tabu-search algorithm in which neighborhoods with different structures are exploited. First, a tabu-search procedure is carried out in which mainly the infeasible region is explored. Once feasibility has been established, a second tabu-search procedure, which analyzes only feasible solutions, is applied. The algorithm has been tested on a wide set of instances. Computational results are discussed.

Improved global–local simulated annealing formulation for solving non-smooth engineering optimization problems

This paper is concerned with a novel optimization algorithm that implements an enhanced formulation of simulated annealing (SA). The new algorithm is denoted as ISA (improved simulated annealing) in the rest of the paper. ISA includes a two-level random search: “global annealing” where all design variables are perturbed simultaneously and “local annealing” where design variables are perturbed one at a time. The improvement with respect to classical SA is in the fact that trial designs are generated always taking care to choose directions along which the cost function may improve. To this purpose, cost function sensitivities are computed in order to properly choose the size of each random perturbation. In addition, the optimization problem is linearized about the current design point if the optimizer ends up in an infeasible region or there is no significant reduction in cost even though the cost function gradient is not close to zero. The linearization is controlled by a trust region model. The optimization algorithm continuously shifts from global to local annealing based on the current best record at the beginning of each cooling cycle. Finally, the cooling schedule is automatically adjusted within ISA based on the convergence behavior. In this work, the ISA algorithm is successfully utilized to solve complicated optimization problems which exhibit non-smooth/non-convex behavior: (i) the large-scale (200 design variables and 3500 constraints) weight minimization of a 200 bar truss under five independent loading conditions; (ii) the configuration optimization of a cantilevered bar truss with 45 elements and 81 design variables; (iii) an example of reverse engineering where in-plane elastic properties of an eight-ply woven composite laminate are to be determined. The performance of ISA is compared to that of classical SA, gradient based optimization codes recently published in literature and commercial software. The results obtained in this study indicate that ISA is a very efficient optimization code. In fact, ISA was much faster than classical SA. The present code allowed about 300 kg weight saving in the 200 bar truss case and about 80 kg in the cantilevered bar truss case. In addition, the residual error on elastic constants in the material identification problem was less than 3%.

Exploiting Tabu Search Memory in Constrained Problems

This paper puts forth a general method to optimize constrained problems effectively when using tabu search. An adaptive penalty approach is used that exploits the short-term memory structure of the tabu list along with the long-term memory of the search results. It is shown to be effective on a variety of combinatorial problems with different degrees and numbers of constraints. The approach requires few parameters, is robust to their setting, and encourages search in promising regions of the feasible and infeasible regions before converging to a final feasible solution. The method is tested on three diverse NP-hard problems, facility layout, system reliability optimization, and orienteering, and is compared with two other penalty approaches developed explicitly for tabu search. The proposed memory-based approach shows consistent strong performance.

A very large-scale neighborhood search algorithm for the multi-resource generalized assignment problem

We propose a metaheuristic algorithm for the multi-resource generalized assignment problem (MRGAP). MRGAP is a generalization of the generalized assignment problem, which is one of the representative combinatorial optimization problems known to be NP-hard. The algorithm features a very large-scale neighborhood search, which is a mechanism of conducting the search with complex and powerful moves, where the resulting neighborhood is efficiently searched via the improvement graph. We also incorporate an adaptive mechanism for adjusting search parameters, to maintain a balance between visits to feasible and infeasible regions. Computational comparisons on benchmark instances show that the method is effective, especially for types D and E instances, which are known to be quite difficult.

An Ejection Chain Approach for the Generalized Assignment Problem

We propose a tabu search algorithm for the generalized assignment problem, which is one of the representative combinatorial optimization problems known to be NP-hard. The algorithm features an ejection chain approach, which is embedded in a neighborhood construction to create more complex and powerful moves. We also incorporate an adaptive mechanism for adjusting search parameters, to maintain a balance between visits to feasible and infeasible regions. Computational results on benchmark instances of small sizes show that the method obtains solutions that are optimal or that deviate by at most 0.16% from the best known solutions. Comparisons with other approaches from the literature show that, for instances of larger sizes, our method obtains the best solutions among all heuristics tested.

Modelling Feasible Design Regions Using Lattice‐based Kernel Methods

AbstractReliability, robustness and general performance are key indicators when attempting to improve the design of existing engineering systems. The search for an improved design is conducted in the space of all possible designs, referred to as the design space. Often when searching this space, infeasible or unsafe regions are encountered, perhaps because particular combinations of parameter values are not physically possible. Such regions need to be avoided during optimization to ensure that any new design is itself feasible. In addition, when considering the reliability and robustness of a design, it may be important to know how far a feasible design is from any boundary. Indeed, it may be the case that such distances are directly related to the reliability and robustness of the design and it may, therefore, be useful to model the boundary between feasible and infeasible regions. This paper proposes the use of Hilbert bases as a method of identifying points in the design space that are close to, or on, the feasible boundary and then uses this information to model the boundary by defining positive definite kernels on new spaces such as hyperspheres. In this way it is possible to model whole surfaces as opposed to single functions. Copyright © 2004 John Wiley & Sons, Ltd.