Bartłomiej Jacek Kubica

Interval methods for computing the Pareto-front of a multicriterial problem

Interval methods are known to be a precise and robust tool of global optimization. Several interval algorithms have been developed to deal with various kinds of this problem. Far less has been written about the use of interval methods in multicriterial optimization. The paper surveys two methods presented by other researchers and proposes a modified approach, combining PICPA algorithm with the use of derivative information. Preliminary numerical results are presented.

Tuning the multithreaded interval method for solving underdetermined systems of nonlinear equations

The paper presents the authors investigations in the field of improving the performance of a multithreaded interval solver of equations systems, targeted for underdetermined systems. New heuristics to choose the interval Newton operator variant and bisection direction are proposed. Numerical results for several benchmark problems are presented and analyzed. Also some other tools and future possible improvements are suggested.

An interval method for seeking the Nash equilibria of non-cooperative games

Computing Nash equilibria in continuous games is a difficult problem. In contrast to discrete games, algorithms developed for continues ones are rather inefficient. This paper proposes a new approach - making use of interval methods we try to solve the problem directly, seeking points that fulfill Nash conditions. We also consider a shared-memory parallelization of the proposed algorithm. Preliminary numerical results are presented. Some new practical aspects of interval methods are considered.

An Interval Global Optimization Algorithm Combining Symbolic Rewriting and Componentwise Newton Method Applied to Control a Class of Queueing Systems

This paper considers two main topics.The first one is a new interval global optimization algorithm, using some symbolic transformations of the optimality conditions. The theory of Groebner bases and the idea of componentwise interval Newton method are used.The second topic is the description of an optimization problem connected with access control to a computer server. This optimization problem is solved by using a new algorithm and, for comparison, by using a classical interval branch-and-bound algorithm.

Presentation of a highly tuned multithreaded interval solver for underdetermined and well-determined nonlinear systems

The paper summarizes author’s investigations in tuning a multithreaded interval branch-and-prune algorithm for nonlinear systems and presents the developed solver. New results for using the box-consistency enforcing operator and a new variant of the initial exclusion phase are presented. Also, a new heuristic to choose the coordinate for bisection is considered. Extensive numerical experiments are analyzed to provide the satisfying version of the algorithm.

Using Quadratic Approximations in an Interval Method for Solving Underdetermined and Well-Determined Nonlinear Systems

This paper considers quadratic approximation as a narrowing tool in an interval branch-and-prune method. We seek the roots of such an approximate equation – a quadratic equation with interval parameters. Heuristics to decide, when to use the developed operator, are proposed. Numerical results for some benchmark problems are presented and analyzed.

Parallel Approach to Monte Carlo Simulation for Option Price Sensitivities Using the Adjoint and Interval Analysis

This paper concerns a new approach to evaluation of Option Price sensitivities using the Monte Carlo simulation, based on the parallel GPU architecture and Automatic Differentiation methods. In order to study rounding errors, the interval arithmetic is used. Considerations are based on two implementations of the algorithm – the sequential and parallel ones. For efficient differentiation, the Adjoint method is employed. Computational experiments include analysis of performance, uncertainty error and rounding error and consider Black-Scholes and Heston models.

Applying an interval method for a four agent economy analysis

This paper presents an application of interval methods for simulation of a policy game played by four main agents of economy: profit maximizing companies active on competitive market, the monopolistic trade union (aggregated wage setters), the government and the central bank. After reduction of companies as an active agent, the resulting three agents decision system is modeled as a Stackelberg game with the central bank as the leader and trade union and government as composite followers, aggregated by means of the Nash solution. The simulation of policy game is provided. A previously developed interval method is applied to find Nash points rigorously.

Preliminary Experiments with an Interval Model-Predictive-Control Solver

Model-Predictive Control (MPC) is a popular advanced control technique, known for its robustness and simplicity in taking control constraints into account. In recent years, the interest grows in applying interval methods to compute MPC. The paper applies interval methods in a simple case. Numerical results for a benchmark problem are presented.

Introduction to the HPC interval methods minisymposium

Interval methods are an approach to performing numerical computations of various kinds, in the presence of errors and uncertainty.