Derivative-free Trust-region Algorithm Research Articles

About the Performance of a Calculus-Based Approach to Building Model Functions in a Derivative-Free Trust-Region Algorithm

This paper examines a calculus-based approach to building model functions in a derivative-free algorithm. This calculus-based approach can be used when the objective function considered is defined via more than one blackbox. Two versions of a derivative-free trust-region method are implemented. The first version builds model functions by using a calculus-based approach, and the second version builds model functions by directly considering the objective function. The numerical experiments demonstrate that the calculus-based approach provides better results in most situations and significantly better results in specific situations.

Derivative-free trust region optimization for robust well control under geological uncertainty

A Derivative-Free Trust-Region (DFTR) algorithm is proposed to solve the robust well control optimization problem under geological uncertainty. Derivative-Free (DF) methods are often a practical alternative when gradients are not available or are unreliable due to cost function discontinuities, e.g., caused by enforcement of simulation-based constraints. However, the effectiveness of DF methods for solving realistic cases is heavily dependent on an efficient sampling strategy since cost function calculations often involve time-consuming reservoir simulations. The DFTR algorithm samples the cost function space around an incumbent solution and builds a quadratic polynomial model, valid within a bounded region (the trust-region). A minimization of the quadratic model guides the method in its search for descent. Because of the curvature information provided by the model-based routine, the trust-region approach is able to conduct a more efficient search compared to other sampling methods, e.g., direct-search approaches. DFTR is implemented within FieldOpt, an open-source framework for field development optimization, and is tested in the Olympus benchmark against two other types of methods commonly applied to production optimization: a direct-search (Asynchronous Parallel Pattern Search) and a population-based (Particle Swarm Optimization). Current results show that DFTR has improved performance compared to the model-free approaches. In particular, the method presented improved convergence, being capable to reach solutions with higher NPV requiring comparatively fewer iterations. This feature can be particularly attractive for practitioners who seek ways to improve production strategies while using an ensemble of full-fledged models, where good convergence properties are even more relevant.

A derivative-free trust-region algorithm with copula-based models for probability maximization problems

This work presents a derivative-free trust-region algorithm for probability maximization problems. We assume that the probability function is continuously differentiable with Lipschitz continuous gradient, but no derivatives are available. The algorithm explores the particular structure of the probability objective function through models based on copulæ. Under reasonable assumptions, the global convergence of the algorithm is analyzed: we prove that all accumulation points of the sequence generated by the algorithm are stationary. The proposed approach is validated by encouraging numerical results on academic and industrial problems.

A new nonmonotone line-search trust-region approach for nonlinear systems

This paper introduces a new derivative-free trust-region algorithm for solving nonlinear systems, based on a new nonmonotone technique and an adaptive radius strategy. It is shown that we can generate the small (large) steps and radii in the cases where iterations are near (far away from) the optimizer. Such a nonmonotone strategy is embedded into the trust region framework and Armijo line search to face with problems which have the narrow curved valley. To prevent resolving the trust-region subproblem, the nonmonotone Armijo line search is used whenever iterations are unsuccessful. In each iteration, the adaptive radius strategy is constructed based on the norm of the best function values. The global and q-quadratic rate of convergence of the new algorithm is proved. Computational results are reported.

A Derivative-Free Trust Region Algorithm with Nonmonotone Filter Technique for Bound Constrained Optimization

We propose a derivative-free trust region algorithm with a nonmonotone filter technique for bound constrained optimization. The derivative-free strategy is applied for special minimization functions in which derivatives are not all available. A nonmonotone filter technique ensures not only the trust region feature but also the global convergence under reasonable assumptions. Numerical experiments demonstrate that the new algorithm is effective for bound constrained optimization. Locally, optimal parameters with respect to overall computational time on a set of test problems are identified. The performance of the best choice of parameter values obtained by the algorithm we presented which differs from traditionally used values indicates that the algorithm proposed in this paper has a certain advantage for the nondifferentiable optimization problems.

A progressive barrier derivative-free trust-region algorithm for constrained optimization

We study derivative-free constrained optimization problems and propose a trust-region method that builds linear or quadratic models around the best feasible and around the best infeasible solutions found so far. These models are optimized within a trust region, and the progressive barrier methodology handles the constraints by progressively pushing the infeasible solutions toward the feasible domain. Computational experiments on 40 smooth constrained problems indicate that the proposed method is competitive with COBYLA, and experiments on two nonsmooth multidisciplinary optimization problems from mechanical engineering show that it can be competitive with the NOMAD software.

ASTRO-DF: A Class of Adaptive Sampling Trust-Region Algorithms for Derivative-Free Stochastic Optimization

We consider unconstrained optimization problems where only “stochastic” estimates of the objective function are observable as replicates from a Monte Carlo oracle. The Monte Carlo oracle is assumed to provide no direct observations of the function gradient. We present ASTRO-DF---a class of derivative-free trust-region algorithms, where a stochastic local model is constructed, optimized, and updated iteratively. Function estimation and model construction within ASTRO-DF is adaptive in the sense that the extent of Monte Carlo sampling is determined by continuously monitoring and balancing measures of sampling error (or variance) and structural error (or model bias) within ASTRO-DF. Such balancing of errors is designed to ensure that Monte Carlo effort within ASTRO-DF is sensitive to algorithm trajectory: sampling is higher whenever an iterate is inferred to be close to a critical point and lower when far away. We demonstrate the almost sure convergence of ASTRO-DF's iterates to first-order critical points when using stochastic polynomial interpolation models. The question of using more complicated models, e.g., regression or stochastic kriging, in combination with adaptive sampling is worth further investigation and will benefit from the methods of proof presented here. We speculate that ASTRO-DF's iterates achieve the canonical Monte Carlo convergence rate, although a proof remains elusive.

A Derivative-Free Trust-Region Algorithm for the Optimization of Functions Smoothed via Gaussian Convolution Using Adaptive Multiple Importance Sampling

In this paper we consider the optimization of a functional $F$ defined as the convolution of a function $f$ with a Gaussian kernel. We propose this type of objective function for the optimization of the output of complex computational simulations, which often present some form of deterministic noise and need to be smoothed for the results to be meaningful. We introduce a derivative-free algorithm that computes trial points from the minimization of a regression model of the noisy function $f$ over a trust region. The regression model is constructed from function values at sample points that are chosen randomly around iterates and trial points of the algorithm. The weights given to the individual sample points in the regression problem are obtained according to an adaptive multiple importance sampling strategy. This has two advantages. First, it makes it possible to reuse all noisy function values collected over the course of the optimization. Second, the resulting regression model converges to the second-o...

On the construction of quadratic models for derivative-free trust-region algorithms

We consider derivative-free trust-region algorithms based on sampling approaches for convex constrained problems and discuss two conditions on the quadratic models for ensuring their global convergence. The first condition requires the poisedness of the sample sets, as usual in this context, while the other one is related to the error between the model and the objective function at the sample points. Although the second condition trivially holds if the model is constructed by polynomial interpolation, since in this case the model coincides with the objective function at the sample set, we show that it also holds for models constructed by support vector regression. These two conditions imply that the error between the gradient of the trust-region model and the objective function is of the order of δ k , where δ k controls the diameter of the sample set. This allows proving the global convergence of a trust-region algorithm that uses two radii, δ k and the trust-region radius. Preliminary numerical experiments are presented for minimizing functions with and without noise.

A derivative-free trust-region algorithm for reliability-based optimization

In this note, we present a derivative-free trust-region (TR) algorithm for reliability based optimization (RBO) problems. The proposed algorithm consists of solving a set of subproblems, in which simple surrogate models of the reliability constraints are constructed and used in solving the subproblems. Taking advantage of the special structure of the RBO problems, we employ a sample reweighting method to evaluate the failure probabilities, which constructs the surrogate for the reliability constraints by performing only a single full reliability evaluation in each iteration. With numerical experiments, we illustrate that the proposed algorithm is competitive against existing methods.

A Computationally Efficient Simulation-Based Optimization Algorithm for Large-Scale Urban Transportation Problems

This paper proposes a computationally efficient simulation-based optimization (SO) algorithm suitable to address large-scale generally constrained urban transportation problems. The algorithm is based on a novel metamodel formulation. We embed the metamodel within a derivative-free trust region algorithm and evaluate the performance of this SO approach considering tight computational budgets. We address a network-wide traffic signal control problem using a calibrated microscopic simulation model of evening peak period traffic of the full city of Lausanne, Switzerland, which consists of more than 600 links and 200 intersections. We control 99 signal phases of 17 intersections distributed throughout the entire network. This SO problem is a high-dimensional nonlinear constrained problem. It is considered large-scale and complex in the fields of derivative-free optimization, traffic signal optimization, and simulation-based optimization. We compare the performance of the proposed metamodel method to that of a traditional metamodel method and that of a widely used commercial signal control software. The proposed method systematically and efficiently identifies signal plans with improved average city-wide travel times.

Derivative-free methods applied to daily production optimization of gas-lifted oil fields

In the oil industry, computer simulators are routinely applied on day to day operations and in what-if analyses. With the increasing complexity of operations, engineers are relying on simulation to synthesize models for mathematical optimization or else applying derivative-free methods for simulation-based optimization, which requires only the sampling of the objective function. This work adapts an algorithm based on augmented Lagrangian and derivative-free trust-region algorithms to handle hard constraints found in production optimization. The effectiveness of the proposed method is assessed in an oil production network and compared with mathematical optimization based on piecewise-linear models.

A derivative-free trust-region algorithm for composite nonsmooth optimization

The derivative-free trust-region algorithm proposed by Conn et al. (SIAM J Optim 20:387–415, 2009) is adapted to the problem of minimizing a composite function $$\varPhi (x)=f(x)+h(c(x))$$ , where $$f$$ and $$c$$ are smooth, and $$h$$ is convex but may be nonsmooth. Under certain conditions, global convergence and a function-evaluation complexity bound are proved. The complexity result is specialized to the case when the derivative-free algorithm is applied to solve equality-constrained problems. Preliminary numerical results with minimax problems are also reported.

A globally convergent trust-region algorithm for unconstrained derivative-free optimization

In this work we propose a derivative-free trust-region algorithm for unconstrained optimization. The algorithm is based on the ideas of Powell (Comput Optim Appl 53:527–555, 2012). We turn his general ideas into a step-form algorithm that avoids unnecessary reductions of the trust-region radius. The results related to the global convergence of the algorithm are presented in detail. We prove that, under reasonable hypotheses, any accumulation point of the sequence generated by the algorithm is stationary.

A Simulation-Based Optimization Framework for Urban Transportation Problems

This paper proposes a simulation-based optimization (SO) method that enables the efficient use of complex stochastic urban traffic simulators to address various transportation problems. It presents a metamodel that integrates information from a simulator with an analytical queueing network model. The proposed metamodel combines a general-purpose component (a quadratic polynomial), which provides a detailed local approximation, with a physical component (the analytical queueing network model), which provides tractable analytical and global information. This combination leads to an SO framework that is computationally efficient and suitable for complex problems with very tight computational budgets. We integrate this metamodel within a derivative-free trust region algorithm. We evaluate the performance of this method considering a traffic signal control problem for the Swiss city of Lausanne, different demand scenarios, and tight computational budgets. The method leads to well-performing signal plans. It leads to reduced, as well as more reliable, average travel times.

On convergence analysis of a derivative-free trust region algorithm for constrained optimization with separable structure

In this paper, we propose a derivative-free trust region algorithm for constrained minimization problems with separable structure, where derivatives of the objective function are not available and can not be directly approximated. At each iteration, we construct a quadratic interpolation model of the objective function around the current iterate. The new iterates are generated by minimizing the augmented Lagrangian function of this model over the trust region. The filter technique is used to ensure the feasibility and optimality of the iterative sequence. Global convergence of the proposed algorithm is proved under some suitable assumptions.

Global Convergence of Radial Basis Function Trust-Region Algorithms for Derivative-Free Optimization

We analyze globally convergent, derivative-free trust-region algorithms relying on radial basis function interpolation models. Our results extend the recent work of Conn, Scheinberg, and Vicente [SIAM J. Optim., 20 (2009), pp. 387--415] to fully linear models that have a nonlinear term. We characterize the types of radial basis functions that fit in our analysis and thus show global convergence to first-order critical points for the ORBIT algorithm of Wild, Regis, and Shoemaker [SIAM J. Sci. Comput., 30 (2008), pp. 3197--3219]. Using ORBIT, we present numerical results for different types of radial basis functions on a series of test problems. We also demonstrate the use of ORBIT in finding local minima on a computationally expensive environmental engineering problem involving remediation of contaminated groundwater.

Global Convergence of Radial Basis Function Trust Region Derivative-Free Algorithms

We analyze globally convergent derivative-free trust region algorithms relying on radial basis function interpolation models. Our results extend the recent work of Conn, Scheinberg, and Vicente [SIAM J. Optim., 20 (2009), pp. 387–415] to fully linear models that have a nonlinear term. We characterize the types of radial basis functions that fit in our analysis and thus show global convergence to first-order critical points for the ORBIT algorithm of Wild, Regis, and Shoemaker [SIAM J. Sci. Comput., 30 (2008), pp. 3197–3219]. Using ORBIT, we present numerical results for different types of radial basis functions on a series of test problems. We also demonstrate the use of ORBIT in finding local minima on a computationally expensive environmental engineering problem.

Parameterize the Center of the Trust Region for Solving Quadratic Interpolation Models

The derivative free trust region algorithm was considered for solving the unconstrained optimization problems. This paper introduces a novel methodology that modified the center of the trust region in order to improve the search region. The main idea is parameterizing the center of the trust region based on the ideas of multi-directional search and simplex search algorithms. The scope of the new region was so expanded by introducing a parameter as to we can find a better descent directions. Experimental results reveal that the new method is more effective than the classic trust region method on the testing problems.

A derivative-free trust-region algorithm for unconstrained optimization with controlled error

In this paper, we consider the unconstrained optimization problem under the following conditions:(S1) The objective function is evaluated with a certain bounded error,(S2) the error is controllable, that is, the objective function can be evaluated to any desired accuracy,and (S3) more accurate evaluation requires a greater computation time.This situation arises in many fields such as engineering and financial problems, where objective function values are obtained from considerable numerical calculation or a simulation.Under (S2) and (S3), it seems reasonable to set the accuracy of the evaluation to be low at points far from a solution, and high at points in the neighborhood of a solution.In this paper, we propose a derivative-free trust-region algorithm based on this idea.For this purpose, we consider (i) how to construct a quadratic model function by exploiting pointwise errors and (ii) how to control the accuracy of function evaluations to reduce the total computation time of the algorithm.For (i), we propose a method based on support vector regression.For (ii), we present an updating formula of the accuracy of which is related to the trust-region radius.We present numerical results for several test problems taken from CUTEr and a financial problem of estimating implied volatilities from option prices.