G. K. Kamenev

Hybrid adaptive methods for approximating a nonconvex multidimensional Pareto frontier

New hybrid methods for approximating the Pareto frontier of the feasible set of criteria vectors in nonlinear multicriteria optimization problems with nonconvex Pareto frontiers are considered. Since the approximation of the Pareto frontier is an ill-posed problem, the methods are based on approximating the Edgeworth-Pareto hull (EPH), i.e., the maximum set having the same Pareto frontier as the original feasible set of criteria vectors. The EPH approximation also makes it possible to visualize the Pareto frontier and to estimate the quality of the approximation. In the methods proposed, the statistical estimation of the quality of the current EPH approximation is combined with its improvement based on a combination of random search, local optimization, adaptive compression of the search region, and genetic algorithms.

Integration of Two Multiobjective Optimization Methods for Nonlinear Problems

In this paper, we bring together two existing methods for solving multiobjective optimization problems described by nonlinear mathematical models and create methods that benefit from both heir strengths. We use the Feasible Goals Method and the NIMBUS method to form new hybrid approaches. The Feasible Goals Method (FGM) is a graphic decision support tool that combines ideas of goal programming and multiobjective methods. It is based on the transformation of numerical information given by mathematical models into a variety of feasible criterion vectors (that is, feasible goals). Visual interactive display of this variety provides information about the problem that helps the decision maker to detect the limits of what is possible. Then, the decision maker can identify a preferred feasible criterion vector on the graphic display. NIMBUS is an interactive multiobjective optimization method capable of solving nonlinear and even nondifferentiable and nonconvex problems. The decision maker can iteratively evaluate the problem to be solved and express personal preferences in a simple form: the method is based on the classification of the criteria, where the decision maker can indicate what kind of changes to the current solution are desirable. We describe two possible hybrids of the FGM and the NIMBUS method for helping in finding the most preferable decision (using simple questions posed to the decision maker). First, feasible criterion values are explored, and the decision makers preferences are expressed roughly in the form of a preferable feasible goal (FGM stage). Then, the identified goal is refined using the classification of the criteria (NIMBUS stage). Alternatively, the two methods can be used interactively. Both the hybrid approaches are here illustrated with an example.

Iterative method for constructing coverings of the multidimensional unit sphere

The stepwise-supplement-of-a-covering (SSC) method is described and examined. The method is intended for the numerical construction of near optimal coverings of the multidimensional unit sphere by neighborhoods of a finite number of points (covering basis). Coverings of the unit sphere are used, for example, in nonadaptive polyhedral approximation of multidimensional convex compact bodies based on the evaluation of their support function for directions defined by points of the covering basis. The SSC method is used to iteratively construct a sequence of coverings, each differing from the previous one by a single new point included in the covering basis. Although such coverings are not optimal, it is theoretically shown that they are asymptotically suboptimal. By applying an experimental analysis, the asymptotic efficiency of the SSC method is estimated and the method is shown to be relatively efficient for a small number of points in the covering basis.

The initial convergence rate of adaptive methods for polyhedral approximation of convex bodies

The convergence rate at the initial stage is analyzed for a previously proposed class of asymptotically optimal adaptive methods for polyhedral approximation of convex bodies. Based on the results, the initial convergence rate of these methods can be evaluated for arbitrary bodies (including the case of polyhedral approximation of polytopes) and the resources sufficient for achieving optimal asymptotic properties can be estimated.

Study of convergence rate and efficiency of two-phase methods for approximating the Edgeworth-Pareto hull

The convergence rate and efficiency of two-phase methods for approximating the Edgeworth-Pareto hull in nonlinear multicriteria optimization problems is studied. A feature of two-phase methods is that the criteria images of randomly generated points of the decision space approach the Pareto frontier via local optimization of adaptively chosen convolutions of criteria. It is shown that the convergence rate of two-phase methods is determined by the metric properties of the set of local extrema of criteria convolutions, specifically, by its upper metric dimension. The efficiency of two-phase methods is examined; i.e., they are compared with hypothetical optimal methods of the same class. It is shown that the efficiency of two-phase methods is determined by the ratio of the ɛ-entropy and ɛ-capacity for the set of local extrema of criteria convolutions.

Optimal growth order of the number of vertices and facets in the class of Hausdorff methods for polyhedral approximation of convex bodies

The internal polyhedral approximation of convex compact bodies with twice continuously differentiable boundaries and positive principal curvatures is considered. The growth of the number of facets in the class of Hausdorff adaptive methods of internal polyhedral approximation that are asymptotically optimal in the growth order of the number of vertices in approximating polytopes is studied. It is shown that the growth order of the number of facets is optimal together with the order growth of the number of vertices. Explicit expressions for the constants in the corresponding bounds are obtained.

Study of an adaptive single-phase method for approximating the multidimensional Pareto frontier in nonlinear systems

AbstractThe problem of approximating the Pareto frontier (nondominated frontier) of the feasible set of criteria vectors in nonlinear multicriteria optimization problems is considered. The problem is solved by approximating the Edgeworth-Pareto hull (EPH), i.e., the maximum set with the same Pareto frontier as the original feasible set of criteria vectors. An EPH approximation method is studied that is based on the statistical accuracy estimation of the current approximation and on adaptive supplement of a metric net whose EPH approximates the desired set. The convergence of the method is proved, estimates for the convergence rate are obtained, and the efficiency of the method is studied in the case of a compact feasible set and continuous criteria functions. It is shown that the convergence rate of the method with respect to the number k of iterations is no lower than

Method for polyhedral approximation of a ball with an optimal order of growth of the facet structure cardinality

o\left( {k^{{1 \mathord{\left/ {\vphantom {1 {\overline {dm} Y}}} \right. \kern-\nulldelimiterspace} {\overline {dm} Y}}} } \right)