A small polygon is a convex polygon of unit diameter. We are interested in small polygons which have the largest area for a given number of vertices n. Many instances are already solved in the ...literature, namely for all odd n, and for n = 4, 6 and 8. Thus, for even n greater than or equal to 10, instances of this problem remain open. Finding those largest small polygons can be formulated as nonconvex quadratic programming problems which can challenge state-of-theart global optimization algorithms. We show that a recently developed technique for global polynomial optimization, based on a semidefinite programming approach to the generalized problem of moments and implemented in the public-domain Matlab package GloptiPoly, can successfully find largest small polygons for n = 10 and n = 12. Therefore this significantly improves existing results in the domain. When coupled with accurate convex conic solvers, GloptiPoly can provide numerical guarantees of global optimality, as well as rigorous guarantees relying on interval arithmetic. Keywords Extremal convex polygons * Global optimization * Nonconvex quadratic programming * Semidefinite programming
This second volume presents research in the field of the mathematical model operation of economic systems, again using as a basis the theory and methods of vector optimization. This volume includes ...three chapters.The first chapter deals with issues related to the theory of the company, modeling and decision-making, while the second deals with issues related to modeling and decision-making in market systems. The third chapter deals with issues related to modeling, forecasting and decision-making.
In solving certain optimization problems, the corresponding Lagrangian dual problem is often solved simply because in these problems the dual problem is easier to solve than the original primal ...problem. Another reason for their solution is the implication of the weak duality theorem which suggests that under certain conditions the optimal dual function value is smaller than or equal to the optimal primal objective value. The dual problem is a special case of a bilevel programming problem involving Lagrange multipliers as upper-level variables and decision variables as lower-level variables. Another interesting aspect of dual problems is that both lower and upper-level optimization problems involve only box constraints and no other equality of inequality constraints. In this paper, we propose a coevolutionary dual optimization (CEDO) algorithm for co-evolving two populations--one involving Lagrange multipliers and other involving decision variables--to find the dual solution. On 11 test problems taken from the optimization literature, we demonstrate the efficacy of CEDO algorithm by comparing it with a couple of nested smooth and nonsmooth algorithms and a couple of previously suggested coevolutionary algorithms. The performance of CEDO algorithm is also compared with two classical methods involving nonsmooth (bundle) optimization methods. As a by-product, we analyze the test problems to find their associated duality gap and classify them into three categories having zero, finite or infinite duality gaps. The development of a coevolutionary approach, revealing the presence or absence of duality gap in a number of commonly-used test problems, and efficacy of the proposed coevolutionary algorithm compared to usual nested smooth and nonsmooth algorithms and other existing coevolutionary approaches remain as the hallmark of the current study. Keywords Dual problem * Duality gap * Optimization * Evolutionary algorithms * Nonsmooth optimization algorithms * Coevolutionary algorithm
This open access book serves as a compact source of information on sine cosine algorithm (SCA) and a foundation for developing and advancing SCA and its applications. SCA is an easy, user-friendly, ...and strong candidate in the field of metaheuristics algorithms. Despite being a relatively new metaheuristic algorithm, it has achieved widespread acceptance among researchers due to its easy implementation and robust optimization capabilities. Its effectiveness and advantages have been demonstrated in various applications ranging from machine learning, engineering design, and wireless sensor network to environmental modeling. The book provides a comprehensive account of the SCA, including details of the underlying ideas, the modified versions, various applications, and a working MATLAB code for the basic SCA.
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