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We propose the general Filter-based Stochastic Algorithm (FbSA) for the global optimization of nonconvex and nonsmooth constrained problems. Under certain conditions on the probability distributions that generate the sample points, almost sure convergence is proved. In order to optimize problems with computationally expensive black-box objective functions, we develop the FbSA-RBF algorithm based on the general FbSA and assisted by Radial Basis Function (RBF) surrogate models to approximate the objective function. At each iteration, the resulting algorithm constructs/updates a surrogate model of the objective function and generates trial points using a dynamic coordinate search strategy similar to the one used in the Dynamically Dimensioned Search method. To identify a promising best trial point, a non-dominance concept based on the values of the surrogate model and the constraint violation at the trial points is used. Theoretical results concerning the sufficient conditions for the almost surely convergence of the algorithm are presented. Preliminary numerical experiments show that the FbSA-RBF is competitive when compared with other known methods in the literature. [ABSTRACT FROM AUTHOR]

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, ). 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. [ABSTRACT FROM AUTHOR]

We describe two algorithms for solving differentiable convex optimization problems constrained to simple sets in ℝn, i.e., sets on which it is easy to project an arbitrary point. The algorithms are optimal in the sense that they achieve an absolute precision of ε in relation to the optimal value in O (1√ε) iterations using only first-order information. This complexity depends on an (unknown) Lipschitz constant L* for the function derivatives and on a (known) strong convexity constant μ* ≥ 0 . The algorithms are extensions of well-known methods devised by Nesterov [Introductory Lectures on Convex Optimization, Kluwer Academic, Boston, 2004], without the need for estimates of L* and including (in the second algorithm) line searches and an adaptive procedure for estimating a strong convexity constant. The algorithms are such that all computed points are feasible, and the complexity analysis follows a simple geometric approach. Numerical tests for box-constrained quadratic problems are presented. [ABSTRACT FROM AUTHOR]

We modify the first order algorithm for convex programming described by Nesterov in his book (in Introductory lectures on convex optimization. A basic course. Kluwer, Boston, ). In his algorithms, Nesterov makes explicit use of a Lipschitz constant L for the function gradient, which is either assumed known (Nesterov in Introductory lectures on convex optimization. A basic course. Kluwer, Boston, ), or is estimated by an adaptive procedure (Nesterov ). We eliminate the use of L at the cost of an extra imprecise line search, and obtain an algorithm which keeps the optimal complexity properties and also inherit the global convergence properties of the steepest descent method for general continuously differentiable optimization. Besides this, we develop an adaptive procedure for estimating a strong convexity constant for the function. Numerical tests for a limited set of toy problems show an improvement in performance when compared with the original Nesterov's algorithms. [ABSTRACT FROM AUTHOR]

In Gonzaga et al. [A globally convergent filter method for nonlinear programming, SIAM J. Optimiz. 14 (2003), pp. 646-669] we discuss general conditions to ensure global convergence of inexact restoration filter algorithms for non-linear programming. In this article we show how to avoid the Maratos effect by means of a second-order correction. The algorithms are based on feasibility and optimality phases, which can be either independent or not. The optimality phase differs from the original one only when a full Newton step for the tangential minimization of the Lagrangian is efficient but not acceptable by the filter method. In this case a second-order corrector step tries to produce an acceptable point keeping the efficiency of the rejected step. The resulting point is tested by trust region criteria. Under the usual hypotheses, the algorithm inherits the quadratic convergence properties of the feasibility and optimality phases. This article includes a comparison between classical Sequential Quadratic Programming (SQP) and Inexact Restoration (IR) iterations, showing that both methods share the same asymptotic convergence properties. [ABSTRACT FROM AUTHOR]

For solving nonsmooth convex constrained optimization problems, we propose an algorithm which combines the ideas of the proximal bundle methods with the filter strategy for evaluating candidate points. The resulting algorithm inherits some attractive features from both approaches. On the one hand, it allows effective control of the size of quadratic programming subproblems via the compression and aggregation techniques of proximal bundle methods. On the other hand, the filter criterion for accepting a candidate point as the new iterate is sometimes easier to satisfy than the usual descent condition in bundle methods. Some encouraging preliminary computational results are also reported. [ABSTRACT FROM AUTHOR]

We present a general filter algorithm that allows a great deal of freedom in the step computation. Each iteration of the algorithm consists basically in computing a point which is not forbidden by the filter, from the current point. We prove its global convergence, assuming that the step must be efficient, in the sense that, near a feasible nonstationary point, the reduction of the objective function is "large." We show that this condition is reasonable, by presenting two classical ways of performing the step which satisfy it. In the first one, the step is obtained by the inexact restoration method of Martínez and Pilotta. In the second, the step is computed by sequential quadratic programming. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we present a general algorithm for nonlinear programming which uses a slanting filter criterion for accepting the new iterates. Independently of how these iterates are computed, we prove that all accumulation points of the sequence generated by the algorithm are feasible. Computing the new iterates by the inexact restoration method, we prove stationarity of all accumulation points of the sequence. [Copyright &y& Elsevier]

This paper presents some examples of ill-behaved central paths in convex optimization. Some contain infinitely many fixed length central segments; others manifest oscillations with infinite variation. These central paths can be encountered even for infinitely differentiable data. [ABSTRACT FROM AUTHOR]

In this paper we present a filter algorithm for nonlinear programming and prove its global convergence to stationary points. Each iteration is composed of a feasibility phase, which reduces a measure of infeasibility and an optimality phase, which reduces the objective function in a tangential approximation of the feasible set. These two phases are totally independent,and the only coupling between them is provided by the filter. The method is independent of the internal algorithms used in each iteration, as long as these algorithms satisfy reasonable assumptions on their efficiency. Under standard hypotheses, we show two results: for a filter with minimum size, the algorithm generates a stationary accumulation point; for a slightly larger filter, all accumulation points are stationary. [ABSTRACT FROM AUTHOR]

The long-term operation of hydro-thermal power generation systems is modeled by a large-scale stochastic optimization problem that includes nonlinear constraints due to the head computation in hydroelectric plants. We do a detailed development of the problem model and state it by a non-anticipative scenario analysis, leading to a large-scale nonlinear programming problem. This is solved by a filter algorithm with sequential quadratic programming iterations that minimize quadratic Lagrangian approximations using exact hessians in L ∞ trust regions. The method is applied to the long-term planning of the Brazilian system, with over 100 hydroelectric and 50 thermoelectric plants, distributed in 5 interconnected subsystems. This problem with 50 synthetically generated inflow scenarios and a horizon of 60 months, amounting to about one million variables and 15000 nonlinear constraints was solved by the filter algorithm in a standard 2016 notebook computer in 10 h of CPU. [ABSTRACT FROM AUTHOR]