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.
Cloud computing alludes to the on-demand availability of personal computer framework resources, primarily information storage and processing power, without the customer's direct personal involvement. ...Cloud computing has developed dramatically among many organizations due to its benefits such as cost savings, resource pooling, broad network access, and ease of management; nonetheless, security has been a major concern. Researchers have proposed several cryptographic methods to offer cloud data security; however, their execution times are linear and longer. A Security Key 4 Optimization Algorithm (SK4OA) with a non-linear run time is proposed in this paper. The secret key of SK4OA determines the run time rather than the size of the data as such is able to transmit large volumes of data with minimal bandwidth and able to resist security attacks like brute force since its execution timings are unpredictable. A data set from Kaggle was used to determine the algorithm's mean and standard deviation after thirty (30) times of execution. Data sizes of 3KB, 5KB, 8KB, 12KB, and 16 KB were used in this study. There was an empirical analysis done against RC4, Salsa20, and Chacha20 based on encryption time, decryption time, throughput and memory utilization. The analysis showed that SK4OA generated lowest mean non-linear run time of 5.545±2.785 when 16KB of data was executed. Additionally, SK4OA's standard deviation was greater, indicating that the observed data varied far from the mean. However, RC4, Salsa20, and Chacha20 showed smaller standard deviations making them more clustered around the mean resulting in predictable run times.
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DOBA, IZUM, KILJ, NUK, PILJ, PNG, SAZU, SIK, UILJ, UKNU, UL, UM, UPUK
We study the out-of-sample properties of robust empirical optimization problems with smooth phi-divergence penalties and smooth concave objective functions, and we develop a theory for data-driven ...calibration of the nonnegative "robustness parameter" delta that controls the size of the deviations from the nominal model. Building on the intuition that robust optimization reduces the sensitivity of the expected reward to errors in the model by controlling the spread of the reward distribution, we show that the first-order benefit of "little bit of robustness" (i.e., delta small, positive) is a significant reduction in the variance of the out-of-sample reward, whereas the corresponding impact on the mean is almost an order of magnitude smaller. One implication is that substantial variance (sensitivity) reduction is possible at little cost if the robustness parameter is properly calibrated. To this end, we introduce the notion of a robust mean-variance frontier to select the robustness parameter and show that it can be approximated using resampling methods such as the bootstrap. Our examples show that robust solutions resulting from "open-loop" calibration methods (e.g., selecting a 90% confidence level regardless of the data and objective function) can be very conservative out of sample, whereas those corresponding to the robustness parameter that optimizes an estimate of the out-of-sample expected reward (e.g., via the bootstrap) with no regard for the variance are often insufficiently robust.
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