Nonlinear Constrained Problem Research Articles

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.

OPTIMIZATION OF HEAT EXCHANGERS FOR GEOTHERMAL DISTRICT HEATING

This paper analyzes the optimal configuration and operating parameters of a heat exchanger in a geothermal district heating system. An optimization algorithm is presented for the nonlinear constrained problem to maximize the annual net profit for a system of counter-flow heat exchangers. Several parameters that affect the net profit are examined, including the mass flow rates of working fluids and heat transfer area, which both directly affect the outgoing temperatures. The performance of the heat exchanger and fuel savings by reducing fuel consumption to generate heat are modeled within the problem formulation. Also, power input to the pump for fluid circulation is included. By formulating these multiple parameters over a wide range of design conditions, the algorithm presents a useful new design tool for the improvement of heat exchanger networks in geothermal systems.

Solving nonlinearly constrained global optimization problem via an auxiliary function method

This paper considers the nonlinearly constrained continuous global minimization problem. Based on the idea of the penalty function method, an auxiliary function, which has approximately the same global minimizers as the original problem, is constructed. An algorithm is developed to minimize the auxiliary function to find an approximate constrained global minimizer of the constrained global minimization problem. The algorithm can escape from the previously converged local minimizers, and can converge to an approximate global minimizer of the problem asymptotically with probability one. Numerical experiments show that it is better than some other well known recent methods for constrained global minimization problems.

Discrete dynamic convexized method for nonlinearly constrained nonlinear integer programming

This paper considers the nonlinearly constrained nonlinear integer programming problem over a bounded box. An auxiliary function is constructed based on a penalty function. By increasing the value of a parameter, minimization of the function by a discrete local search method can escape successfully from a previously converged discrete local minimizer. An algorithm is designed based on minimizing the auxiliary function with increasing values of the parameter. Numerical experiments show that the algorithm is robust and efficient.

Remarks on a benchmark nonlinear constrained optimization problem

Remarks on a benchmark nonlinear constrained optimization problem are made. Due to a citation error, two absolutely different results for the benchmark problem are obtained by independent researchers. Parallel simulated annealing using simplex method is employed in our study to solve the benchmark nonlinear constrained problem with mistaken formula and the best-known solution is obtained, whose optimality is testified by the Kuhn-Tucker conditions.

An [formula omitted]-approximation method in nonlinear vector optimization

In this paper, a new approach for a solution of a nonlinear multiobjective programming problem is introduced. An equivalent η -approximated vector optimization problem is constructed by a modification of the objective and the constraint functions in the original multiobjective programming problem. The connection between (weak) efficient points in the original multiobjective programming problem and its equivalent η -approximated vector optimization problem is proved. In this way, optimality conditions for nonlinear constrained multiobjective programming problems having invex and/or generalized invex objective and constraint functions (with respect to the same functions η ) are obtained.

Constrained Optimization by the α Constrained Particle Swarm Optimizer

In this study, α constrained particle swarm optimizer αPSO, which is the combination of the α constrained method and particle swarm optimization, is proposed to solve constrained optimization problems. The α constrained methods can convert algorithms for unconstrained problems to algorithms for constrained problems using the α level comparison, which compares the search points based on the satisfaction level of constraints. In the αPSO, the agents who satisfy the constraints move to optimize the objective function and the agents who don't satisfy the constraints move to satisfy the constraints. The effectiveness of the αPSO is shown by comparing the αPSO with GENOCOP5.0, and other PSO-based methods on some nonlinear constrained problems.

Numerical methods for large-scale nonlinear optimization

Recent developments in numerical methods for solving large differentiable nonlinear optimization problems are reviewed. State-of-the-art algorithms for solving unconstrained, bound-constrained, linearly constrained and non-linearly constrained problems are discussed. As well as important conceptual advances and theoretical aspects, emphasis is also placed on more practical issues, such as software availability.

A Constrained Nonlinear Inversion Approach to Quantitative Interpretation of Self-potential Anomalies Caused by Cylinders, Spheres and Sheet-like Structures

An interpretative method based on a nonlinearly mathematical optimization concept has been developed in this paper, in order to interpret self-potential anomalies (SP) due to horizontal cylinder, vertical cylinder, sphere and sheet-like structures. This interpretative method comprises three main steps. The first step is to formulate mathematically a nonlinearly constrained minimization problem (NCMP) to describe the geophysical problem related to the studied structure. The second one is to suggest an interior penalty function in order to convert the nonlinearly constrained minimization problem (NCMP) into a nonlinearly unconstrained minimization one (NUMP). The third step is to solve the converted nonlinearly unconstrained minimization problem (NUMP) by the well-known Hooke and Jeeves direct search algorithm in order to estimate the geophysical parameters of the studied structure, i.e., depth, polarization angle, electric dipole moment (magnitude of polarization) and geometric shape factor. The Hooke and Jeeves direct search algorithm is purposely chosen for being robust and its application to SP data allows a rapid convergence towards the optimal estimate of parameters. This interpretative method was first tested on theoretical synthetic models with different random noise, where a very close agreement was obtained between assumed and evaluated parameters.

SIGEST

Previous article Next article Full AccessSIGESThttps://doi.org/10.1137/SIREAD000047000001000097000001BibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractDonald Knuth, the renowned computer scientist and mathematician, commented in a recent interview [D. E. Knuth, "All questions answered," Notices of the American Mathematical Society, 29 (2002), pp.~318--324] that "...my main conclusion after spending ten years of my life on the TeX project is that software is hard. It's harder than anything else I've ever had to do." Everyone who has written software for general-purpose consumption will agree with these statements. Software is the most frequent transformation through which applied mathematics becomes useful. For all but the simplest problems, software is hard because it must not only correctly implement one or more theoretical algorithms, but also cope gracefully with finite-precision arithmetic, poor scaling, problems that fail to satisfy the properties required for the theory to apply, and the vagaries of users who ignore or knowingly violate guidelines specified by the author. Writing software is hard, and writing well about software is equally hard. However, this issue's SIGEST paper, "SNOPT: An SQP Algorithm for Large-Scale Constrained Optimization," by Philip Gill, Walter Murray, and Michael Saunders, proves that it is possible to write a clear, interesting, and lively paper about a complicated algorithm and the associated software. SNOPT represents a long accumulation of varied theoretical advances and hard-earned experiences with real-world problems, and also exemplifies Knuth's assertion. Solving nonlinearly constrained optimization problems via a sequence of quadratic programming (QP) subproblems was first proposed more than 40 years ago, and a 1977 paper by a prominent optimization researcher stated that a sequential quadratic programming (SQP) method could be "programmed in an afternoon." However, this early optimism about the ease of SQP software development turned out to be unjustified for many reasons, in part because of the special nature of the QP subproblems and the importance of fine details in the linear algebra calculations, particularly for large-scale problems. The SNOPT paper includes a concise overview not just of SQP methods, but also of the nonobvious decisions to be made and the tricky problems to be solved. For example, should the software attempt to determine when the specified nonlinear constraints are likely to be infeasible (i.e., there is no point that satisfies them)? If so, given that there is no general procedure guaranteed to detect infeasibility, what can be done that is efficient yet has a high probability of correctness? The list of this and other questions that must be answered when solving constrained optimization problems is very long, and the paper provides the thoughtful answers embodied in SNOPT. The world of constrained optimization software---including the development of methods and the collection of test problems---has noticeably changed since the SNOPT paper first appeared in the SIAM Journal on Optimization in 2002 (volume 12, pp. 979--1006). The authors have accordingly updated the paper's discussion of alternative software (notably, several interior-point methods). In addition, they have expended considerable effort (and machine cycles!) to apply the latest version of SNOPT to the most recent versions of the two most popular sets of test problems, whose largest instances have increased significantly in dimension since 2002. We are grateful to the authors for giving SIAM Review readers a paper that explains the details of the theory and implementation of SQP methods, along with strategies for addressing problems arising along the way that are provably impossible, but that users want to solve anyway. Previous article Next article FiguresRelatedReferencesCited ByDetails Trajectory Optimization Based on Interval AnalysisAIAA Guidance, Navigation, and Control Conference | 14 June 2012 Cross Ref Volume 47, Issue 1| 2005SIAM Review History Published online:04 August 2006 Article & Publication DataArticle DOI:10.1137/SIREAD000047000001000097000001Article page range:pp. 97-97ISSN (print):0036-1445ISSN (online):1095-7200Publisher:Society for Industrial and Applied Mathematics

Augmented Lagrangian Algorithms Based on the Spectral Projected Gradient Method for Solving Nonlinear Programming Problems

The spectral projected gradient method SPG is an algorithm for large-scale bound-constrained optimization introduced recently by Birgin, Martinez, and Raydan. It is based on the Raydan unconstrained generalization of the Barzilai-Borwein method for quadratics. The SPG algorithm turned out to be surprisingly effective for solving many large-scale minimization problems with box constraints. Therefore, it is natural to test its perfomance for solving the sub-problems that appear in nonlinear programming methods based on augmented Lagrangians. In this work, augmented Lagrangian methods which use SPG as the underlying convex-constraint solver are introduced (ALSPG) and the methods are tested in two sets of problems. First, a meaningful subset of large-scale nonlinearly constrained problems of the CUTE collection is solved and compared with the perfomance of LANCELOT. Second, a family of location problems in the minimax formulation is solved against the package FFSQP.

Interior point algorithms for nonlinear constrained least squares problems

We consider Nonlinear Least Squares problems with equality and inequality constraints and propose a numerical technique that integrates methods for unconstrained problems, based on Gauss–Newton algorithm, with FAIPA, the Feasible Arc Interior Point Algorithm for constrained optimization. We also present some numerical results on test problems available in the literature and compare them with the quasi-Newton version of FAIPA. We also describe an application to the identification of mechanical parameters of composite materials. The present algorithms are globally convergent, very robust and efficient.

Interdigitation for effective design space exploration using iSIGHT

Optimization studies for nonlinear constrained problems (i.e. most complex engineering design problems) have repeatedly shown that (i) no single optimization technique performs best for all design problems, and (ii) in most cases, a mix of techniques perform better than a single technique for a given design problem. iSIGHT TM is a generic software framework for integration, automation, and optimization of design processes that has been developed on the foundation of interdigitation: the strategy of combining multiple optimization algorithms to exploit their desirable aspects for solving complex problems. With the recent paradigm shift from traditional optimization to design space exploration for evaluating “what-if” scenarios and trade-off studies, iSIGHT has grown from an optimization software system to a complete design exploration environment, providing a suite of design exploration tools including a collection of optimization techniques, design of experiments techniques, approximation methods, and probabilistic quality engineering methods. Likewise, the interdigitation design methodology embodied in iSIGHT has grown to support the interdigitation of all design exploration tools for effective design space exploration. In this paper we present an overview of iSIGHT, past and present, of the interdigitation design methodology and its implementation for multiple design exploration tools, and of an industrial case study for which elements of this methodology have been applied.

Mathematical modelling for keyhole surgery simulations: a biomechanical model for spleen tissue

There is growing interest in the potential for computer simulations to provide good training materials for abdominal endoscopic or 'keyhole' surgery. This paper describes a preliminary study of the mechanical properties of abdominal (spleen) tissue. We show how experimental force-displacement data can be used to derive a biomechanical model for the tissue as an incompressible, homogeneous, isotropic nonlinear elastic material with an exponential stress-strain law. We also show how the model can be used to predict the response of the tissue to a surgical probe. This involves solving a complicated nonlinear constrained boundary-value problem, and there is a good fit between the computed solution and experimental data.

A Conic Trust-Region Method for Nonlinearly Constrained Optimization *

Trust-region methods are powerful optimization methods. The conic model method is a new type of method with more information available at each iteration than standard quadratic-based methods. Can we combine their advantages to form a more powerful method for constrained optimization? In this paper we give a positive answer and present a conic trust-region algorithm for non-linearly constrained optimization problems. The trust-region subproblem of our method is to minimize a conic function subject to the linearized constraints and the trust region bound. The use of conic functions allows the model to interpolate function values and gradient values of the Lagrange function at both the current point and previous iterate point. Since conic functions are the extension of quadratic functions, they approximate general nonlinear functions better than quadratic functions. At the same time, the new algorithm possesses robust global properties. In this paper we establish the global convergence of the new algorithm under standard conditions.

A neural-based nonlinear L 1-norm optimization algorithm for diagnosis of networks

Based on exact penalty function, a new neural network for solving the L1-norm optimization problem is proposed. In comparison with Kennedy and Chua’s network(1988), it has better properties.Based on Bandler’s fault location method(1982), a new nonlinearly constrained L1-norm problem is developed. It can be solved with less computing time through only one optimization processing. The proposed neural network can be used to solve the analog diagnosis L1 problem. The validity of the proposed neural networks and the fault location L1 method are illustrated by extensive computer simulations.

Pseudospectral Legendre-based optimal computation of nonlinear constrained variational problems

It is well known that, spectrally accurate solution can be maintained if the grids on which a nonlinear physical problem is to be solved must be obtained by spectrally accurate techniques. In this paper, the pseudospectral Legendre method for general nonlinear smooth and nonsmooth constrained problems of the calculus of variations is studied. The technique is based on spectral collocation methods in which the trajectory, x( t), is approximated by the Nth degree interpolating polynomial, using Legendre-Gauss-Lobatto points as the collocation points, and Lagrange polynomials as trial functions. The integral involved in the formulation of the problem is approximated based on Legendre-Gauss-Lobatto integration rule, thereby reducing the problem to a nonlinear programming one to which existing well-developed algorithms may be applied. The method is easy to implement and yields very accurate results. Illustrative examples are included to confirm the convergence of the pseudospectral Legendre method. Moreover, a numerical experiment (on a nonsmooth problem) indicates that by applying a smoothing filter procedure to the pseudospectral Legendre approximation, one can recover the nonsmooth solution within spectral accuracy.

A CONJUGATE PROJECTION VXRIABLE METRIC ALGORITHM FOR LINEAR AND NONLINEAR CONSTRAINED OPTIMIZMION PROBLEM

The gradient projection method is a class of effective algorithms for constrained optimization problem,and it has an important position in optimization. But the projection used in gradient projection method is an orthogonal projection which does not contain the second derivative information of objective and constrained functions,thus the convergence rate of gradient projection method is not contented. This paper introduces a kind of conjugateprojection with which a conjugate projection variable metric algorithm for linear and nonlinear constrained optiinization problem is presented. The global convergence of the algorithm is proved under certain conditions.Since.the conjugate projection in algorithm finely containsthe second derivative information of objective and constrained functions,the convergence rate of algorithm in the paper is expected to be accelerated.The numerical results illustrate that the algorithm in the paper is more effective than gradient projection method.

Methods for finding feasible points in constrained optimization

Methods for finding feasible points are needed in some numerical optimization algorithms to initiate the search for a local minimum. Other methods perform more efficiently if a nearly feasible starting point is used. In addition, a feasible solution may be acceptable as the final design in some practical applications. Therefore, constraint correction methods are quite useful. Such methods based on penalty functions and primal approach are described and discussed. To gain insight into the methods a two-variable problem is used to analyze the methods. Several structural design problems are used to study the numerical behavior of the methods and compare their performance. It is concluded that, in general, the primal methods are more efficient than the penalty function methods. Also, no method has been found that is guaranteed to find a feasible point for general constrained nonlinear problems starting from an arbitrary point. In case a method fails, random points should be used to restart the procedure to search for a feasible point.

Optimality conditions for state constrained nonlinear control problems

The problem of optimal control of nonlinear control and state constrained control problems, where the state constraint may involve differential operators and the cost functionals may be nonsmooth, is studied. For this class of problems, necessary optimality conditions using techniques from infinite dimensional optimization theory adapted to the framework of control problems are derived. It is shown that the underlying structure admits a considerable relaxation of the classical constraint qualifications. The theory then is applied to examples of various nonlinear elliptic equations and state constraints.