Abstract not found

A novel global optimization method based on an Augmented Lagrangian framework is introduced for continuous constrained nonlinear optimization problems. At each outer iteration k the method requires the εk-global minimization of the Augmented Lagrangian with simple constraints, where εk → ε. Global convergence to an ε-global minimizer of the original problem is proved. The subproblems are solved using the αBB method. Numerical experiments are presented.

Abstract: Integrated scientific-technical computing (ISTC) environments play an increasing role in advanced systems modeling and optimization. MathOptimizer Professional (MOP) has been recently developed to solve nonlinear optimization problems formulated in the ISTC system Mathematica. We introduce this software package, and review its key functionality and options. MOP is then used to solve illustrative circle packing problems, including both well-frequented models and a new (more difficult) model-class.

We consider the problem of sampling a point from an arbitrary distribution π over an arbitrary subset S of an integer hyper-rectangle. Neither the distribution π nor the support set S are assumed to be available as explicit mathematical equations but may only be defined through oracles and in particular computer programs. This problem commonly occurs in black-box discrete optimization as well as counting and estimation problems. The generality of this setting and high-dimensionality of S precludes the application of conventional random variable generation methods. As a result, we turn to Markov Chain Monte Carlo (MCMC) sampling, where we execute an ergodic Markov chain that converges to π so that the distribution of the point delivered after sufficiently many steps can be made arbitrarily close to π. Unfortunately, classical Markov chains such as the nearest neighbor random walk or the co-ordinate direction random walk fail to converge to π as they can get trapped in isolated regions of the support set. To surmount this difficulty, we propose Discrete Hit-and-Run (DHR), a Markov chain motivated by the Hit-and-Run algorithm known to be the most efficient method for sampling from log-concave distributions over convex bodies in Rn. We prove that the limiting distribution of DHR is π as desired, thus enabling us to sample approximately from π by delivering the last iterate of a sufficiently

We propose a risk-neutral second best toll pricing scheme to account for the possible nonuniqueness of user equilibrium solutions. The scheme is designed to optimize for the expected objective value as the UE solution varies within the solution set. We show that such a risk-neutral scheme can be formulated as a stochastic program, which complements the traditional risk-prone second best toll pricing (SBTP) approach and the risk-averse SBTP approach we developed recently. The proposed model can be solved by a simulation-based optimization algorithm that contains three major steps: characterization of the UE solution set, random sampling over the solution set, and a two-phase simulation optimization step. Numerical results illustrate that the proposed risk-neutral design scheme is less aggressive than the risk-prone scheme and less conservative than the risk-averse scheme, and may thus be more preferable from a toll designer’s point of view.

We develop new Markov chain Monte Carlo samplers for neighborhood generation in global optimization algorithms based on Hit-and-Run. The success of Hit-and-Run as a sampler on continuous domains motivated Discrete Hit-and-Run with random biwalk for discrete domains. However, the potential in efficiencies in the implementation, which requires a randomization at each move to create the biwalk, lead us to a different approach that uses fixed patterns in generating the biwalks. We define Sphere and Box Biwalks that are pattern-based and easily implemented for discrete and mixed continuous/discrete domains. The pattern-based Hit-and-Run Markov chains preserve the convergence properties of Hitand-Run to a target distribution. They also converge to continuous Hit-and-Run as the mesh of the discretized variables becomes finer, approaching a continuum. Moreover, we provide bounds on the finite time performance for the discrete cases of Sphere and Box Biwalks. We embed our samplers in an Improving Hit-and-Run global optimization algorithm and test their performance on a number of global optimization test problems.

In a recent paper, Birgin, Floudas and Martínez introduced an augmented Lagrangian method for global optimization. In their approach, augmented Lagrangian subproblems are solved using the αBB method and convergence to global minimizers was obtained assuming feasibility of the original problem. In the present research, the algorithm mentioned above will be improved in several crucial aspects. On the one hand, feasibility of the problem will not be required. Possible infeasibility will be detected in finite time by the new algorithms and optimal infeasibility results will be proved. On the other hand, finite termination results that guarantee optimality and/or feasibility up to any required precision will be provided. An adaptive modification in which subproblem tolerances depend on current feasibility and complementarity will also be given. The adaptive algorithm allows the augmented Lagrangian subproblems to be solved without requiring unnecessary potentially high precisions in the intermediate steps of the method, which improves the overall efficiency. Experiments showing how the new algorithms and results are related to practical computations will be given. Key words: deterministic global optimization, augmented Lagrangians, nonlinear programming, algorithms, numerical experiments. 1