Narrow escape and narrow capture problems which describe the average times required to stop the motion of a randomly travelling particle within a domain have applications in various areas of science. While for general domains, it is known how the escape time decreases with the increase of the trap sizes, for some specific 2D and 3D domains, higher-order asymptotic formulas have been established, providing the dependence of the escape time on the sizes and locations of the traps. Such results allow the use of global optimisation to seek trap arrangements that minimise average escape times. In a recent paper (Iyaniwura (2021) SIAM Rev. 63(3), 525–555), an explicit size- and trap location-dependent expansion of the average mean first passage time (MFPT) in a 2D elliptic domain was derived. The goal of this work is to systematically seek global minima of MFPT for $1\leq N\leq 50$ traps in elliptic domains using global optimisation techniques and compare the corresponding putative optimal trap arrangements for different values of the domain eccentricity. Further, an asymptotic formula for the average MFPT in elliptic domains with N circular traps of arbitrary sizes is derived, and sample optimal configurations involving non-equal traps are computed.

Computational methods and a program to obtain crystal structures that have the perfectly identical diffraction patterns, i.e. structure factors with the same absolute values and the same lattice symmetry are discussed. This is directly related to the uniqueness of solutions in crystal structure determination of single-crystal/powder-crystal samples from diffraction data. In order to solve the problem, it is necessary to solve a system of quadratic equations. The framework of positive-semidefinite programming is used herein to solve the system efficiently.

This paper presents a novel motion planning approach inspired by the Dynamic Programming (DP) applicable to multi degree of freedom robots (mobile or stationary) and autonomous vehicles. The proposed discrete–time algorithm enables a robot to reach its destination through an arbitrary obstacle field in the fewest number of time steps possible while minimizing a secondary objective function. Furthermore, the resulting optimal trajectory is guaranteed to be globally optimal while incorporating state constraints such as velocity, acceleration, and jerk limits. The optimal trajectories furnished by the algorithm may be further updated in real time to accommodate changes in the obstacle field and/or cost function. The algorithm is proven to terminate in a finite number of steps without its computational complexity increasing with the type or number of obstacles. The effectiveness of the global and replanning algorithms are demonstrated on a planar mobile robot with three degrees of freedom subject to velocity and acceleration limits. The computational complexity of the two algorithms are also compared to that of an A*–type search.

A flexible apple sorting hand-claw with three fingers evenly on the spiral chuck is designed. Each finger with series double hinges is driven by one air-cylinder. Sizes of hand-claw and parameters of two torsion springs round axes of series double hinges are obtained via global optimization based on the grid method so that the two torsion springs exhibit a coordinative function to enhance compliance for grasping apples of different sizes under the same system pressure. Besides, the critical pressures and bending deflection between different working processes are studied. Results show that hinge 2 begins to rotate at 0.0270 MPa pressure, and when the pressure increases to 0.0634 MPa and 0.0746 MPa, hand-claw begins to touch the biggest and the smallest apples respectively. The curve of torsional angle versus pressure presents that angular velocity droops with homogeneous extension. Therefore, it is able to reduce impact and time in vain during the grasping process. Crucially, the grasping forces for grasping the smallest and the biggest apple are 4 N and 4.38 N respectively under the same 0.5 MPa pressure, indicating the excellent adaption performance and robust result for size variation of apples without any force sensors.

In this paper the tip-over stability of mobile robots during manipulation with redundant arms is investigated in real-time. A new fast-converging algorithm, called the Circles Of INitialization (COIN), is proposed to calculate globally optimal postures of redundant serial manipulators. The algorithm is capable of trajectory following, redundancy resolution, and tip-over prevention for mobile robots during eccentric manipulation tasks. The proposed algorithm employs a priori training data generated from an exhaustive resolution of the arm's redundancy along a single direction in the manipulator's workspace. This data is shown to provide educated initial guess that enables COIN to swiftly converge to the global optimum for any other task in the workspace. Simulations demonstrate the capabilities of COIN, and further highlight its convergence speed relative to existing global search algorithms.

A new way of incorporating powder diffraction data into a cost function to predict the crystalline structure of inorganic solids is proposed. This approach was applied to the following series of compounds: cubic SrTiO3, tetragonal NaNbO3, TiO2 (anatase), tetragonal CaTiO3, and hexagonal BaTiO3. A tremendous increase in the efficiency of obtaining the correct structure is achieved when a cost function based upon this new approach is applied to these problems.

In this paper, a new hybrid simulated annealing algorithm for constrained globaloptimization is proposed. We have developed a stochastic algorithm called ASAPSPSA thatuses Adaptive Simulated Annealing algorithm (ASA). ASA is a series of modifications to thebasic simulated annealing algorithm (SA) that gives the region containing the globalsolution of an objective function. In addition, Simultaneous Perturbation StochasticApproximation (SPSA) method, for solving unconstrained optimization problems, is used torefine the solution. We also propose Penalty SPSA (PSPSA) for solving constrainedoptimization problems. The constraints are handled using exterior point penalty functions.The combination of both techniques ASA and PSPSA provides a powerful hybrid optimizationmethod. The proposed method has a good balance between exploration and exploitation withvery fast computation speed, its performance as a viable large scale optimization methodis demonstrated by testing it on a number of benchmark functions with 2 - 500 dimensions.In addition, applicability of the algorithm on structural design was tested and successfulresults were obtained

Facility location problems are one of the most common applications of optimization methods. Continuous formulations are usually more accurate, but often result in complex problems that cannot be solved using traditional optimization methods. This paper examines theuse of a global optimization method—AGOP—for solving location problems where the objective function is discontinuous. This approach is motivated by a real-world application in wireless networks design.

In this paper we establish necessary as well assufficient conditions for a given feasible point to be a globalminimizer of smooth minimization problems with mixed variables.These problems, for instance, cover box constrained smooth minimizationproblems and bivalent optimization problems. In particular, ourresults provide necessary global optimality conditions for differenceconvex minimization problems, whereas our sufficient conditionsgive easily verifiable conditions for global optimality of variousclasses of nonconvex minimization problems, including the class ofdifference of convex and quadratic minimization problems. Wediscuss numerical examples to illustrate the optimalityconditions

In the last decades, robust estimation has been widely applied to overcome the presence of gross errors in observations. Recently it has been shown that robust estimation, if computed appropriately, is able to cope with a larger amount of gross errors and also capable of providing reliable estimations in situations where systematic errors are present. It is the purpose of this paper to propose the use of global robust estimation for computing baselines strongly affected by multipath effects.

We present below a new series of conjectures and openproblems in the fields of (global) Optimization and Matrix analysis, in thesame spirit as our recently published paper [J.-B. Hiriart-Urruty, Potpourri of conjectures and open questions in Nonlinear analysis and Optimization. SIAMReview49 (2007) 255–273]. With each problem come a succinct presentation, a list of specificreferences, and a view on the state of the art of the subject.

The purpose of this paper is to demonstrate that, for globally minimize one dimensional nonconvex problems withboth twice differentiable function and constraint, we can propose an efficientalgorithm based on Branch and Bound techniques. The method is firstdisplayed in the simple case with an interval constraint. The extension is displayedafterwards to the general case with an additional nonconvex twicedifferentiable constraint. A quadratic bounding function which is betterthan the well known linear underestimator is proposed while w-subdivision is added to support the branching procedure. Computational results on several andvarious types of functions show the efficiency of our algorithms and theirsuperiority with respect to the existing methods.

We present a review of the main “global optimization" methods. The paper comprises one introduction and two parts. In the introduction, we recall some generalities about non linear constraint-less optimization and we list some classifications which have been proposed for the global optimization methods. We then describe, in the first part, various “classical" global optimization methods, most of which available long before the appearance of Simulated Annealing (a key event in this field). There exists plenty of papers and books dealing with these methods, and studying in particular their convergence properties. The second part of the paper is devoted to more recent or atypical methods, mostly issued from combinatorial optimization. The three main methods are “metaheuristics": Simulated Annealing (and derived techniques), Tabu Search and Genetic Algorithms; we also describe three other less known methods. For these methods, theoretical studies of convergence are less abundant in the literature, and the use of convergence results is by far more limited in practice. However, the fitting of some of these techniques to continuous variables problems gave very promising results; that question is not discussed in detail in the paper, but useful references allowing to deepen the subject are given.

This work develops a class of stochastic global optimization algorithms that are Kiefer-Wolfowitz (KW) type procedures with an added perturbing noise and partial step size restarting. The motivation stems from the use of KW-type procedures and Monte Carlo versions of simulated annealing algorithms in a wide range of applications. Using weak convergence approaches, our effort is directed to proving the convergence of the underlying algorithms under general noise processes.

Markovian algorithms for estimating the global maximum or minimum of real valued functions defined on some domain Ω ⊂ ℝd are presented. Conditions on the search schemes that preserve the asymptotic distribution are derived. Global and local search schemes satisfying these conditions are analysed and shown to yield sharper confidence intervals when compared to the i.i.d. case.

This paper is a study of the error in approximating the global maximum of a Brownian motion on the unit interval by observing the value at randomly chosen points. One point of view is to look at the error from random sampling for a given fixed Brownian sample path; another is to look at the error with both the path and observations random. In the first case we show that for almost all Brownian paths the error, normalized by multiplying by the square root of the number of observations, does not converge in distribution, while in the second case the normalized error does converge in distribution. We derive the limiting distribution of the normalized error averaged over all paths.

The stochastic process corresponding to the simulated annealing optimization algorithm is generalized to the case of an arbitrary state space. Conditions for the strong and weak convergence of the process are established. In addition the relation between the size of the generating distributions and the possible rate of cooling is studied.

Simulated annealing is a randomized algorithm which has been proposed for finding globally optimum least-cost configurations in large NP-complete problems with cost functions which may have many local minima. A theoretical analysis of simulated annealing based on its precise model, a time-inhomogeneous Markov chain, is presented. An annealing schedule is given for which the Markov chain is strongly ergodic and the algorithm converges to a global optimum. The finite-time behavior of simulated annealing is also analyzed and a bound obtained on the departure of the probability distribution of the state at finite time from the optimum. This bound gives an estimate of the rate of convergence and insights into the conditions on the annealing schedule which gives optimum performance.