09h45 - 10h10
General Finite Ellipse Packings in an Optimized Circle
A generic ellipse packing problem is defined as follows: find the best non-overlapping arrangement of n given ellipses with (in principle) arbitrary geometry and orientation inside a given type of optimized container set. Here we consider the ellipse packing problem with respect to an optimized container circle which has minimal radius. Following the review of selected topical literature, we introduce a model formulation approach based on using embedded Lagrange multipliers. Our optimization model has been implemented using the computing system Mathematica and solved by the LGO nonlinear (global and local) optimization software package linked to Mathematica. Our study supported by illustrative numerical results demonstrates that the Lagrangian modeling approach combined with nonlinear optimization tools can be successfully applied to very challenging ellipse packing problems.
10h10 - 10h35
DIRECT-type algorithms for constrained global optimization
The well-known DIRECT (DIviding RECTangles) algorithm for global optimization requires bounds on variables and does not naturally address general constraints. In this talk, we will review existing approaches for constrained global optimization that have been developed for using with DIRECT-type algorithms. When the feasible region is defined by linear constraints, it can be efficiently covered by simplices. We will demonstrate this advantage by using simplicial partitioning based Lc-DISIMPL algorithm. For the problems with nonlinear constraints, we will review existing approaches, highlighting pros and cons of each. Then we will propose improvements how to overcome observed disadvantages. We will finish with the results of a computational study showing the impact of introduced speed-up techniques.
10h35 - 11h00
Constrained global optimization algorithms and their numerical testing
Lipschitz multidimensional constrained global optimization problems where both the objective function and constraints can be multiextremal and non-differentiable functions are considered in this contribution. The Strongin’s index approach for solving one-dimensional constrained problems is extended to geometric algorithms basing on efficient diagonal partitions. A class of GKLS-based multidimensional test problems (recently proposed by the authors) with continuously differentiable multiextremal objective functions and nonlinear constraints is also described. Some numerical experiments with the introduced constrained global optimization methods on this class are reported.