In the literature on generalized pattern search algorithms, convergence to a stationary point of a once continuously differentiable cost function is established under the assumption that the cost function can be evaluated exactly. However, there is a large class of engineering problems where the numerical evaluation of the cost function involves the solution of systems of differential algebraic equations. Since the termination criteria of the numerical solvers often depend on the design parameters, computer code for solving these systems usually defines a numerical approximation to the cost function that is discontinuous with respect to the design parameters. Standard generalized pattern search algorithms have been applied heuristically to such problems, but no convergence properties have been stated. In this paper we extend a class of generalized pattern search algorithms to include a subprocedure that adaptively controls the precision of the approximating cost functions. The numerical approximations to the cost function need not define a continuous function. Our algorithms can be used for solving linearly constrained problems with cost functions that are at least locally Lipschitz continuous. Assuming that the cost function is smooth, we prove that our algorithms converge to a stationary point. Under the weaker assumption that the cost function is only locally Lipschitz continuous, we show that our algorithms converge to points at which the Clarke generalized directional derivatives are nonnegative in predefined directions. An important feature of our adaptive precision scheme is the use of coarse approximations in the early iterations, with the approximation precision controlled by a test. We show by numerical experiments that such an approach leads to substantial time savings in minimizing computationally expensive functions.

1 aPolak, Elijah1 aWetter, Michael uhttps://simulationresearch.lbl.gov/publications/precision-control-generalized-pattern01926nas a2200133 4500008004100000245011300041210006900154300001200223490000700235520142300242100002001665700001801685856008901703 2005 eng d00aBuilding design optimization using a convergent pattern search algorithm with adaptive precision simulations0 aBuilding design optimization using a convergent pattern search a a603-6120 v373 aWe propose a simulation–precision control algorithm that can be used with a family of derivative free optimization algorithms to solve optimization problems in which the cost function is defined through the solutions of a coupled system of differential algebraic equations (DAEs). Our optimization algorithms use coarse precision approximations to the solutions of the DAE system in the early iterations and progressively increase the precision as the optimization approaches a solution. Such schemes often yield a significant reduction in computation time. We assume that the cost function is smooth but that it can only be approximated numerically by approximating cost functions that are discontinuous in the design parameters. We show that this situation is typical for many building energy optimization problems.We present a new building energy and daylighting simulation program, which constructs approximations to the cost function that converge uniformly on bounded sets to a smooth function as precision is increased.We prove that for our simulation program, our optimization algorithms construct sequences of iterates with stationary accumulation points. We present numerical experiments in which we minimize the annual energy consumption of an office building for lighting, cooling and heating. In these examples, our precision control algorithm reduces the computation time up to a factor of four.

1 aWetter, Michael1 aPolak, Elijah uhttps://simulationresearch.lbl.gov/publications/building-design-optimization-using-000513nas a2200133 4500008004100000245011300041210006900154260001200223300001200235490000700247100002000254700001800274856008700292 2004 eng d00aBuilding Design Optimization Using a Convergent Pattern Search Algorithm with Adaptive Precision Simulations0 aBuilding Design Optimization Using a Convergent Pattern Search A c09/2004 a603-6120 v371 aWetter, Michael1 aPolak, Elijah uhttps://simulationresearch.lbl.gov/publications/building-design-optimization-using00369nas a2200121 4500008004100000022001500041245003000056210002800086100002000114700001800134700001900152856007600171 2004 eng d aLBNL-5465800aBuildOpt 1.0.1 validation0 aBuildOpt 101 validation1 aWetter, Michael1 aPolak, Elijah1 aCarey, Van, P. uhttps://simulationresearch.lbl.gov/publications/buildopt-101-validation01637nas a2200145 4500008004100000245010400041210006900145260001200214300001200226490000700238520111900245100002001364700001801384856008901402 2004 eng d00aA convergent optimization method using pattern search algorithms with adaptive precision simulation0 aconvergent optimization method using pattern search algorithms w c11/2004 a327-3380 v253 aThermal building simulation programs, such as EnergyPlus, compute numerical approximations to solutions of systems of differential algebraic equations. We show that the exact solutions of these systems are usually smooth in the building design parameters, but that the numerical approximations are usually discontinuous due to adaptive solvers and finite precision computations. If such approximate solutions are used in conjunction with optimization algorithms that depend on smoothness of the cost function, one needs to compute high precision solutions, which can be prohibitively expensive if used for all iterations. For such situations, we have developed an adaptive simulation–precision control algorithm that can be used in conjunction with a family of derivative free optimization algorithms. We present the main ingredients of the composite algorithms, we prove that the resulting composite algorithms construct sequences with stationary accumulation points, and we show by numerical experiments that using coarse approximations in the early iterations can significantly reduce computation time.

1 aWetter, Michael1 aPolak, Elijah uhttps://simulationresearch.lbl.gov/publications/convergent-optimization-method-using01634nas a2200241 4500008004100000245010400041210006900145260002700214300001400241490000800255520083600263653002201099653001801121653002201139653001901161653001701180653003201197100002001229700001801249700002401267700001601291856008501307 2003 eng d00aA convergent optimization method using pattern search algorithms with adaptive precision simulation0 aconvergent optimization method using pattern search algorithms w aEindhoven, Netherlands a1393-14000 vIII3 aIn solving optimization problems for building design and control, the cost function is often evaluated using a detailed building simulation program. These programs contain code features that cause the cost function to be discontinuous. Optimization algorithms that require smoothness can fail on such problems. Evaluating the cost function is often so time-consuming that stochastic optimization algorithms are run using only a few simulations, which decreases the probability of getting close to a minimum. To show how applicable direct search, stochastic, and gradient-based optimization algorithms are for solving such optimization problems, we compare the performance of these algorithms in minimizing cost functions with different smoothness. We also explain what causes the large discontinuities in the cost functions.

10acoordinate search10adirect search10agenetic algorithm10ahooke–jeeves10aoptimization10aparticle swarm optimization1 aWetter, Michael1 aPolak, Elijah1 aAugenbroe, Godfried1 aHensen, Jan uhttps://simulationresearch.lbl.gov/publications/convergent-optimization-method-0