Journal of Global Optimization最新文献

Nonlinear dependence on a probability measure has recently been encountered with increasing intensity in stochastic optimization. This type of dependence corresponds to many situations in applications; it can appear in problems static (one-stage), dynamic with finite (multi-stage) or infinite horizon, and single- and multi-objective ones. Moreover, the nonlinear dependence can appear not only in the objective functions but also in the constraint sets. In this paper, we will consider static one-objective problems in which the nonlinear dependence appears in the objective function and may also appear in the constraint sets. In detail, we consider “deterministic” constraint sets, whose dependence on the probability measure is nonlinear, constraint sets determined by second-order stochastic dominance, and sets given by mean-risk problems. The last mentioned instance means that the constraint set corresponds to solutions which guarantee acceptable values of both criteria. To obtain relevant assertions, we employ the stability results given by the Wasserstein metric, based on the ( {{mathcal {L}}}_{1} ) norm. We mainly focus on the case in which a solution has to be obtained on the basis of the data and of investigating a relationship between the original problem and its empirical version.

A procedure is presented for finding the shortest network connecting three given undirected points, subject to a curvature constraint on both the path joining two of the points and the path that connects to the third point. The problem is a generalisation of the Fermat–Torricelli problem and is related to a shortest curvature-constrained path problem that was solved by Dubins. The procedure has the potential to be applied to the optimal design of decline networks in underground mines.

In this paper, we design a neural network architecture to approximate the weakly efficient frontier of convex vector optimization problems (CVOP) satisfying Slater’s condition. The proposed machine learning methodology provides both an inner and outer approximation of the weakly efficient frontier, as well as an upper bound to the error at each approximated efficient point. In numerical case studies we demonstrate that the proposed algorithm is effectively able to approximate the true weakly efficient frontier of CVOPs. This remains true even for large problems (i.e., many objectives, variables, and constraints) and thus overcoming the curse of dimensionality.

Quadratic programs with box constraints involve minimizing a possibly nonconvex quadratic function subject to lower and upper bounds on each variable. This is a well-known NP-hard problem that frequently arises in various applications. We focus on two convex relaxations, namely the reformulation–linearization technique (RLT) relaxation and the SDP-RLT relaxation obtained by combining the Shor relaxation with the RLT relaxation. Both relaxations yield lower bounds on the optimal value of a quadratic program with box constraints. We show that each component of each vertex of the RLT relaxation lies in the set ({0,frac{1}{2},1}). We present complete algebraic descriptions of the set of instances that admit exact RLT relaxations as well as those that admit exact SDP-RLT relaxations. We show that our descriptions can be converted into algorithms for efficiently constructing instances with (1) exact RLT relaxations, (2) inexact RLT relaxations, (3) exact SDP-RLT relaxations, and (4) exact SDP-RLT but inexact RLT relaxations. Our preliminary computational experiments illustrate that our algorithms are capable of generating computationally challenging instances for state-of-the-art solvers.

We propose a second-order method for unconditional minimization of functions f(z) of complex arguments. We call it the mixed Newton method due to the use of the mixed Wirtinger derivative (frac{partial ^2f}{partial {bar{z}}partial z}) for computation of the search direction, as opposed to the full Hessian (frac{partial ^2f}{partial (z,{bar{z}})^2}) in the classical Newton method. The method has been developed for specific applications in wireless network communications, but its global convergence properties are shown to be superior on a more general class of functions f, namely sums of squares of absolute values of holomorphic functions. In particular, for such objective functions minima are surrounded by attraction basins, while the iterates are repelled from other types of critical points. We provide formulas for the asymptotic convergence rate and show that in the scalar case the method reduces to the well-known complex Newton method for the search of zeros of holomorphic functions. In this case, it exhibits generically fractal global convergence patterns.

We consider multiobjective optimization problems with at least one computationally expensive constraint function and propose a novel surrogate-assisted evolutionary algorithm that can incorporate preference information given a priori. We employ Kriging models to approximate expensive objective and constraint functions, enabling us to introduce a new selection strategy that emphasizes the generation of feasible solutions throughout the optimization process. In our innovative model management, we perform expensive function evaluations to identify feasible solutions that best reflect the decision maker’s preferences provided before the process. To assess the performance of our proposed algorithm, we utilize two distinct parameterless performance indicators and compare them against existing algorithms from the literature using various real-world engineering and benchmark problems. Furthermore, we assemble new algorithms to analyze the effects of the selection strategy and the model management on the performance of the proposed algorithm. The results show that in most cases, our algorithm has a better performance than the assembled algorithms, especially when there is a restricted budget for expensive function evaluations.

Interval-based branch & bound solvers are commonly used for solving Nonlinear Continuous global Optimization Problems (NCOPs). In each iteration, the solver strategically chooses and processes a node within the search tree. The node is bisected and the two generated offspring nodes are processed by filtering methods. For each of these nodes, the solver also searches for new feasible solutions in order to update the best candidate solution. The cost of this solution is used for pruning non-optimal branches of the search tree. Thus, node selection and finding new solutions, stands as pivotal aspects in the functionality of these kind of solvers. The ability to find close-to-optimal solutions early in the search process may discard extensive non-optimal search space regions, thereby effectively reducing the overall size of the search tree. In this work, we propose three novel node selection algorithms that use the feasible solutions obtained through a cost-effective iterative method. Upon updating the best candidate solution, these algorithms strategically choose the node containing this solution for subsequent processing. The newly introduced strategies have been incorporated as node selection methods in a state-of-the-art branch & bound solver, showing promising results in a set of 57 benchmark instances.

We present an edge labeling of order-k Voronoi diagrams, (V_k(S)), of point sets S in the plane, and study properties of the regions defined by them. Among them, we show that (V_k(S)) has a small orientable cycle and path double cover, and we identify configurations that cannot appear in (V_k(S)) for small values of k. This paper also contains a systematic study of well-known and new properties of (V_k(S)), all whose proofs only rely on elementary geometric arguments in the plane. The maybe most comprehensive study of structural properties of (V_k(S)) was done by D.T. Lee (On k-nearest neighbor Voronoi diagrams in the plane) in 1982. Our work reviews and extends the list of properties of higher order Voronoi diagrams.

The aim of this paper is to present dual optimality conditions for the difference of two extended real valued increasing co-radiant functions. We do this by first characterizing dual optimality conditions for the difference of two nonpositive increasing co-radiant functions. Finally, we present dual optimality conditions for the difference of two extended real valued increasing co-radiant functions. Our approach is based on the Toland–Singer formula.

Deterministic direct-search methods have been successfully used to address real-world challenging optimization problems, including the beam angle optimization (BAO) problem in radiation therapy treatment planning. BAO is a highly non-convex optimization problem typically treated as the optimization of an expensive multi-modal black-box function which results in a computationally time consuming procedure. For the recently available modalities of radiation therapy with protons (instead of photons) further efficiency in terms of computational time is required despite the success of the different strategies developed to accelerate BAO approaches. Introducing randomization into otherwise deterministic direct-search approaches has been shown to lead to excellent computational performance, particularly when considering a reduced number (as low as two) of random poll directions at each iteration. In this study several randomized direct-search strategies are tested considering different sets of polling directions. Results obtained using a prostate and a head-and-neck cancer cases confirmed the high-quality results obtained by deterministic direct-search methods. Randomized strategies using a reduced number of polling directions showed difficulties for the higher dimensional search space (head-and-neck) and, despite the excellent mean results for the prostate cancer case, outliers were observed, a result that is often ignored in the literature. While, for general global optimization problems, mean results (or obtaining the global optimum once) might be enough for assessing the performance of the randomized method, in real-world problems one should not disregard the worst-case scenario and beware of the possibility of poor results since, many times, it is only possible to run the optimization problem once. This is even more important in healthcare applications where the mean patient does not exist and the best treatment possible must be assured for every patient.