Minimizing the Lennard-Jones potential, the most-studied modelproblem for molecular conformation, is an unconstrained globaloptimization problem with a large number of local minima. In thispaper, the problem is reformulated as an equality constrainednonlinear programming problem with only linear constraints. Thisformulation allows the solution to approached through infeasibleconfigurations, increasing the basin of attraction of the globalsolution. In this way the likelihood of finding a global minimizeris increased. An algorithm for solving this nonlinear program isdiscussed, and results of numerical tests are presented.

A review of statistical models for global optimization is presented. Rationality of the search for a global minimum is formulated axiomatically and the features of the corresponding algorithm are derived from the axioms. Furthermore the results of some applications of the proposed algorithm are presented and the perspectives of the approach are discussed.

Multidimensional scaling with city block norm in embedding space is considered. Construction of the corresponding algorithm is reduced to minimization of a piecewise quadratic function. The two level algorithm is developed combining combinatorial minimization at upper level with local minimization at lower level. Results of experimental investigation of the efficiency of the proposed algorithm are presented as well as examples of its application to visualization of multidimensional data.

It is known that the problem of minimizing a convex functionf(x) over a compact subsetX of ℝⁿ can be expressed as minimizing max{g(x, y)|y ∈X}, whereg is a support function forf[f(x) ≥g(x, y), for ally ∈X andf(x)=g(x, x)]. Standard outer-approximation theory can then be employed to obtain outer-approximation algorithms with procedures for dropping previous cuts. It is shown here how this methodology can be extended to nonconvex nondifferentiable functions.

Based on some phenomena from human society and nature, we propose a binary affinity genetic algorithm (aGA) by adopting the following strategies: the population is adaptively updated to avoid stagnation; the newly generated individuals will be ensured to survive for some generations in order for them to have time to show their good genes; new individuals and the old ones are balanced to have the advantages of both. In order to quantitatively analyze the selective pressure, the concept of selection degree and a simple linear control equation are introduced. We can maintain the diversity of the evolutionary population by controlling the value of the selection degree. Performance of aGA is further enhanced by incorporating local search strategies.

The detection of gravitational waves is a long-awaited event in modern physics and, to achieve this challenging goal, detectors with high sensitivity are being used or are under development. In order to extract gravitational signals emitted by coalescing binary systems of compact objects (neutron stars and/or black holes), from noisy data obtained by interferometric detectors, the matched filter technique is generally used. Its computational kernel is a box-constrained global optimization problem with many local solutions and a highly nonlinear and expensive objective function, whose derivatives are not available. To tackle this problem, we designed a real-coded genetic algorithm that exploits characteristic features of the problem itself; special attention was devoted to the choice of the initial population and of the recombination operator. Computational experiments showed that our algorithm is able to compute a reasonably accurate solution of the optimization problem, requiring a much smaller number of function evaluations than the grid search, which is generally used to solve this problem. Furthermore, the genetic algorithm largely outperforms other global optimization algorithms on significant instances of the problem.

In this paper, we consider a nonlinear programming problem for which the constraint set may be infeasible. We propose an algorithm based on a large family of augmented Lagrangian functions and analyze its global convergence properties taking into account the possible infeasibility of the problem. We show that, in a finite number of iterations, the algorithm stops detecting the infeasibility of the problem or finds an approximate feasible/optimal solution with any required precision. We illustrate, by means of numerical experiments, that our algorithm is reliable for different Lagrangian/penalty functions proposed in the literature.

We discuss the geometrical interpretation of a well-known smoothing operator applied to the Molecular Distance Geometry Problem (MDGP), and we then describe a heuristic approach based on Variable Neighbourhood Search on the smoothed and original problem. This algorithm often manages to find solutions having higher accuracy than other methods. This is important as small differences in the objective function value may point to completely different 3D molecular structures.

Determining global integer extrema of an real-valued box-constrained multivariate quadratic functions is a very difficult task. In this paper, we present an analytic method, which is based on a combinatorial optimization approach in order to calculate global integer extrema of a real-valued box-constrained multivariate quadratic function, whereby this problem will be proven to be as NP-hard via solving it by a Travelling Salesman instance. Instead, we solve it using eigenvalue theory, which allows us to calculate the eigenvalues of an arbitrary symmetric matrix using Newton’s method, which converges quadratically and in addition yields a Jordan normal form with $$1 \times 1$$ -blocks, from which a special representation of the multivariate quadratic function based on affine linear functions can be derived. Finally, global integer minimizers can be calculated dynamically and efficiently most often in a small amount of time using the Fourier–Motzkin- and a Branch and Bound like Dijkstra-algorithm. As an application, we consider a box-constrained bivariate and multivariate quadratic function with ten arguments.

Computing the global minimum of a potential energy function is very difficult because it typically has a very large number of local minima which may grow exponentially with problem size. In this work we use a deterministic algorithm for finding the global minimum of this function. The algorithm is based on a branch and bound method that uses techniques of interval analysis. Using the Lennard-Jones potential function, the proposed approach was successfully applied to two example problems.