Publicado em CD-ROMIn this paper we investigate the performance of DIRECT algorithm when
solving constrained engineering design problems. For this purpose, the
hyperbolic penalty approach is employed and the algorithm is modified in
order to preserve feasibility of solutions. The algorithm is illustrated on
six well–known engineering problems with promising results. Comparisons
with other global optimization solvers are reported and discussed

Rocha, Ana Maria A. C.

Vilaça, Rita

Universidade do Minho: RepositoriUM

X Congreso Galego de Estat´ıstica e Investigacio´n de Operacio´nsPontevedra, 3–4–5 de novembro de 2011EMBEDDING THE DIRECT ALGORITHM IN A PENALTYAPPROACH FOR SOLVING ENGINEERING DESIGN PROBLEMSRita Vilac¸a1 and Ana Maria A. C. Rocha11University of Minho, School of Engineering, PortugalABSTRACTIn this paper we investigate the performance of DIRECT algorithm whensolving constrained engineering design problems. For this purpose, thehyperbolic penalty approach is employed and the algorithm is modified inorder to preserve feasibility of solutions. The algorithm is illustrated onsix well–known engineering problems with promising results. Comparisonswith other global optimization solvers are reported and discussed.Keywords: global optimization, constrained optimization, DIRECT algorithm,hyperbolic penalty function, engineering design problems.1. INTRODUCTIONMany engineering applications, such as structural optimization, engineering de-sign, VLSI design, economics, allocation and location problems (Floudas and Parda-los 1987), involve difficult optimization problems that must be solved efficiently andeffectively. Due to the nature of these applications, the solutions usually need to beconstrained in specific parts of the search space that are delimited by linear and/ornonlinear constraints.Hence, the kind of problems to be addressed in this paper areminimize f(x)subject to gi(x) ≤ 0x ∈ Ω(1)where f : Rn → R and gi : Rn → Rm, for i = 1, . . . ,m, are nonlinear functions andΩ = {x ∈ Rn : l ≤ x ≤ u} is a closed set. We assume that the objective function f isnonconvex and may possess many local minima in the feasible region. This class ofglobal optimization problems arises frequently in engineering applications.Different deterministic as well as stochastic algorithms have been developed fortackling such problems. Deterministic approaches such as Feasible Direction andGeneralized Gradient Descent make strong assumptions on the continuity and dif-ferentiability of the objective function (Floudas and Pardalos 1987). Therefore their1applicability is limited since these characteristics are rarely met in problems thatarise in real life applications. On the other hand, stochastic optimization algorithmssuch as Genetic Algorithms, Evolution Strategies, Particle Swarm Optimization andElectromagnetism-like mechanism do not make such assumptions and they have beensuccessfully applied for tackling constrained optimization problems during the pastfew years (Hu et al. 2003, Coello 2002, Parsopoulos and Vrahatis 2002, Rocha andFernandes 2009).A well-known approach for solving this kind of constrained optimization prob-lems is based on penalty functions (Coello 2000). The penalty techniques transformthe constrained problem into an unconstrained problem by penalizing f when con-straints are violated and then minimizing the penalty function using methods forunconstrained problems.This paper presents a numerical study of a penalty approach, based on the hy-perbolic penalty function, where the unconstrained problems are solved by a globaldeterministic algorithm, the DIRECT algorithm.In Section 2, the penalty approach is described, as well as the proposed DIRECTalgorithm. Following, the numerical results when applying the proposed method arepresented to some engineering design problems, described in the literature. Finallysome conclusions and final remarks are reported and discussed in Section 4.2. THE PENALTY APPROACHPenalty functions methods have made themselves among the most common meth-ods for solving constrained optimization problems by their advantages of simplicityand easy to be implemented. Hence, in order to solve problem (1) a penalty approachwill be used that converts a constrained optimization problem into an unconstrainedone by adding a penalty term to the objective function of infeasible solutions so thatthey will be penalized for violating the constraints. Although penalty functions arevery simple and easy to implement they often require several parameters that areusually problem dependent and chosen by priori knowledge by users. Too large pa-rameters lead to heavy selective pressure that would cause the algorithm hard toconverge to satisfactory solutions not mention to the global optimum. However toosmall parameters make the search too broad and hard to find a feasible solution.Different penalty functions have been suggested and can also be classified based onthe way the penalties are added: death, static, dynamic, annealing, adaptive (Coello2002).Here, the solution of problem (1) is obtained by solving a sequence of subproblems,whose objective function is given by the hyperbolic penalty method (Xavier 2001)P (x, λ, τ) = f(x) +m∑i=1(λgi(x) +√λ2gi(x)2 + τ2) (2)where λ, τ ≥ 0 and λ → ∞, τ → 0. The sequence of subproblems is obtained bycontrolling the two parameters in two different phases of the optimization process.In the first phase, the initial parameter λ increases, thus causing a reduction in thepenalty to the points outside the feasible region while at the same time there is a2reduction in the penalty for the points inside the feasible region. This phase continuesuntil a feasible point is obtained. From this point on, λ remains constant and thevalues of τ decrease sequentially.In this context, the subproblems to be solved in each k iteration are given byminimizex∈ΩP (x, λk, τk) (3)and are approximately solved using the DIRECT algorithm that is a determinis-tic global optimization method. DIRECT algorithm from Jones et al. (1993), is anacronym for DIviding RECTangles, and it is a deterministic sampling global optimiza-tion method designed for finding the global minima for bound constrained non-smoothproblems, where no derivative information is needed. It is guaranteed to converge tothe global optimal function value, if the objective function is continuous or at leastcontinuous in the neighborhood of a global optimum. The most important advantageof DIRECT stems from its approach to balancing local and global search - the simpleidea of not sampling just one point per iteration, but rather sampling several pointsusing all possible weightings of local versus global search. This approach leads to analgorithm with no tuning parameters, making the algorithm easy-to-use and robust.The first step in the DIRECT algorithm is to transform the search space to bethe unit hypercube. The function is then sampled at the center-point of this cube.Computing the function value at the center-point instead of doing it at the verticesis an advantage when dealing with problems in higher dimensions. The hypercubeis then divided into smaller hyperrectangles whose centerpoints are also sampled.Instead of using a Lipschitz constant when determining the rectangles to samplenext, DIRECT identifies a set of potentially optimal rectangles in each iteration.All potentially optimal rectangles are further divided into smaller rectangles whosecenter-points are sampled. See (Finkel 2003; Finkel and Kelly 2004) for more details.The hyperbolic penalty method for constrained optimization embedding the DI-RECT algorithm is described in the Algorithm 1. Step 4 refers to the solution ofthe subproblem (3) parametrized by the penalty parameters using the DIRECT al-gorithm. The herein used measure of constraint violation is given byviol =m∑i=1max{0, gi(x)}. (4)3. NUMERICAL RESULTSProblems of practical interest are important for assessing the effectiveness of anyalgorithm. Thus, to evaluate the performance of the embedded DIRECT algorithmin the hyperbolic penalty function method a set of 6 benchmark engineering problemsis used.Following there is a summary of the characteristics of the engineering design prob-lems selected, where all of them have simple bounds and inequality constraints (Costaand Fernandes 2008, Lee and Geem 2005, Ray and Liew 2002, Rocha and Fernandes2009).3Algorithm 1 Hyperbolic Penalty + DIRECT Algorithm1. Given λ1 = 20, τ1 = 10, ǫmax = 10−6, λmax = 1012, τmin = 10−12, γλ = 2,(γλ > 1), γτ = 0.01, (0 < γτ < 1), η∗ = 10−6, kmax = 202. Set k = 13. While (viol > ǫmax OR |f∗ − f(xk)| > 10−4|f∗|) AND k < kmax do4. Find an approximate minimizer xk to the subproblem (3) using DIRECT5. If viol > ǫmax then λk+1 = min{λmax, γλλk} and τk+1 = τk6. Else τk+1 = max{τmin, γτ τk} and λk+1 = λk7. Set k = k + 18. End while1. Design of a tension/compression spring where the weight of a tension/ com-pression spring is minimized, subject to constraints on the minimum deflection,shear stress, surge frequency, limits on outside diameter and on design variables.The problem has 3 design variables and 4 inequality constraints.2. Design of a speed reducer where the objective in this problem is to minimizethe total weight of a speed reducer, subject to constraints on bending stress ofthe gear teeth, surface stress, transverse deflections of the shafts and stresses inthe shafts. It has 7 design variables and 11 inequality constraints.3. Design of a disc brake where the objective is to minimize both the mass ofthe brake and the stopping time. The problem has 4 design variables and 6inequality constraints.4. Design of a tubular column where the objective is to minimize the total cost ofthe material and construction of a tubular column. The problem has 2 designvariables and 2 inequality constraints.5. Design of three-bar truss where the objective is to minimize the volume of a3-bar truss structure, subject to stress constraints. The problem has 2 designvariables that represent cross-sectional areas of two bars (the third bar is equalto the first bar), and 3 inequality constraints.6. Design of four-bar truss structure where the objective is to minimize the volumeand the displacement of a 4-bar truss structure, subject to stress constraints.The problem has 4 design variables that represent cross-sectional areas and 1inequality constraint.The results reported in Table 1 include the number of iterations (Nit), the numberof objective function evaluations (Nfe) and the optimal solution found (Fopt) withthe hyperbolic penalty method as described in Algorithm 1. The stopping criteriawas the specified in Step 3 of the algorithm. We remark that we also stopped thealgorithm when the penalty parameters stabilized and the solution could not improve,since DIRECT is a deterministic algorithm. The conditions used to stop the DIRECTalgorithm (in Step 4 of Algorithm 1) were the maximum number of iterations and the4maximum objective function evaluations set to 10000. To assess the performance ofthe hyperbolic penalty method Table 1 also reports the results of the penalty approachwith the quadratic penalty function, when applied to the same engineering problems.Table 1: Comparison results of the penalty approach (in terms of Fopt, Nfe and Nit).spring speed brake tubular 3-bar 4-barf∗ 0.0126 2994.4999 0.1274 26.5313 263.896 1400quadratic Fopt 0.0280 3040.4774 0.1329 26.7402 263.8975 1400.0532penalty Nfe 40115 120375 29163 3619 100155 10056function Nit 4 12 2 11 10 1hyperbolic Fopt 0.0156 2995.9333 0.1274 26.5329 263.9665 1400.0542penalty Nfe 8117 90156 40099 30070 50068 30090function Nit 8 9 7 3 3 3The results reported in Table 1 show that, in general, the hyperbolic penaltyfunction performs better when compared with the quadratic penalty function. Theoptimal solutions found are more closer to the optimal solution known in the literature(f∗). We remark, for only three problems the hyperbolic function is more expensivein terms of objective function evaluations than quadratic function, but in these casesthe optimal solution found is better.4. CONCLUSIONSThis paper presents the hyperbolic penalty method to solve constrained engi-neering design problems. The unconstrained problems that arrive from the penaltyapproach are solved by a deterministic global optimization method, named DIRECT.To asses the performance of the herein proposed method a set of six well–knownengineering problems is solved. A comparison with a quadratic penalty function isreported and show promising results.REFERENCESCoello, C.A.C. (2000) Use of a self-adaptive penalty approach for engineering opti-mization problems, Comput. Ind., 41, 113-127.Coello, C.A.C. (2002) Theoretical and numerical constraint-handling techniques usedwith evolutionary algorithms: a survey of the state of the art, Computer Methods inapplied mechanics and engineering, 191, 1245–1287.Costa, M.F.P. and Fernandes, E.M.G.P. (2008) Efficient solving of engineering de-sign problems by an interior point 3-D filter line search method, Numerical Analysisand Applied Mathematics, T.E. Simos, G. Psihoyios and Ch. Tsitouras (Eds.), AIPConference Proceedings, 1048, 197–200, Springer-Verlag.Finkel, D.E. and Kelley, C.T. (2004) Convergence Analysis of the DIRECT Algo-rithm [Report]. North Carolina State University: Center for Research in ScientificComputation, Raleigh.5Finkel, D.E. (2003) DIRECT Optimization Algorithm User Guide [Report]. Centerfor Research in Scientific Computation: North Carolina State University, Raleigh.Floudas, C.A. and Pardalos, P.M. (1987) A collection of test problems for constrainedglobal optimization algorithms, LNCS, 455. Springer-Verlag.Hu, X., Eberhart, R.C. and Shi, Y. (2003) Engineering optimization with particleswarm. In: Proc. 2003 IEEE Swarm Intelligence Symposium, 53-57.Jones, D.R., Perttunen, C.D. and Stuckman, B.E. (1993) Lipschitzian optimizationwithout the Lipschitz constant. Journal of Optimization Theory and Applications,79(1), 157–181.Lee, K.S. and Geem, Z.W. (2005) A new meta-heuristic algorithm for continuousengineering optimization: Harmony search theory and practice, Comput. MethodsAppl. Mech. Eng., 194, 3902-3933.Parsopoulos, K.E. and Vrahatis, M.N. (2002) Particle swarm optimization methodfor constrained optimization problems. In Sincak et al., eds.: Intelligent Technolo-gies Theory and Application: New Trends in Intelligent Technologies. Volume 76 ofFrontiers in Artificial Intelligence and Applications, 214-220.Ray, T. and Liew, K.M. (2002) A swarm metaphor for multiobjective design opti-mization, Engineering Optimization, 34(2), 141-153.Rocha, A.M.A.C. and Fernandes, E.M.G.P. (2009) Hybridizing the electromagnetism-like algorithm with descent search for solving engineering design problems, Interna-tional Journal of Computer Mathematics, Vol. 86, Issue 10–11, 1932–1946.Xavier, A.E. (2001) Hyperbolic penalty: A new method for nonlinear programmingwith inequalities. International Transactions in Operational Research 8, 659-671.6

Embedding the DIRECT algorithm in a penalty approach for solving engineering design problems

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