Malfatti’s constrained optimization problem

In 1803 Italian mathematician Malfatti posed the following problem how to pack three non-overlapping circles of maximum total area in a given triangle. Malfatti originally assumed that the solution to this problem are three circles inscribed in a triangle such that each circle tangent to other two and touches two sides of the triangle. Now it is well known that Malfatti’s solution is not optimal. The problem for the first time was treated as a global optimization problem in Enkhbat [9]. In this paper, we consider a new formulation of Malfatti’s problem called Malfatti’s constrained optimization problem. The new problem is formulated as a nonconvex optimization problem with nonlinear constraints. Numerical experiments were conducted on Python for the cases. Мальфаттын зааглалтай бодлого Хураангуй: 1803 онд анх Италийн математикч Мальфатт өгөгдсөн гурвалжинд хамгийн их талбайтай, давхцахгүйгээр гурван тойргийг хэрхэн байрлуулах вэ? гэсэн бодлогыг тавьж байсан бөгөөд энэхүү бодлогын шийд нь гурвалжинд багтсан гурван тойргуудын тойрог бүр нөгөө хоёр тойргийг, гурвалжны хоёр талыг шүргэсэн байна гэж үзсэн.Энэ нь хараахан оновчтой шийд биш байсан ба [9] ажилд анх Мальфаттын бодлогыг бодох глобал оновчлолын бодлогыг томьёолж шийдийг олох арга алгоритм боловсруулсан. Энэхүү судалгаанд бид Мальфаттын бодлогын шинэ томьёолол буюу Хөдөлгөөнт Мальфаттын бодлогыг авч үзсэн. Тус бодлого нь шугаман бус хязгаарлалттай гүдгэр бус оновчлолын бодлого юм. Тоон туршилтыг Python дээр хийсэн. Түлхүүр үгс: Мальфаттын бодлого, гүдгэр бус оновчлол, тойрог, гурвалжи

The Connection Between Pareto Optimality and Portfolio Growth Rate

Portfolio optimization plays an important role in investment sciences. We examine the classical Markowitz model from a viewpoint of Pareto optimality. We consider a multi-objective optimization problem by maximizing the return of a portfolio and minimizing risk. We show that for appropriate weights, the Pareto optimal solution of the multi-objective optimization is a solution to the problem of maximizing a portfolio growth rate. Numerical results were provided using Mathlab. Багцын Өгөөжийн Өсөлт ба Парето Оновчлол Хураангуй: Хөрөнгө оруулалтын шинжлэх ухаанд багцын оновчлол чухал үүрэг гүйцэтгэдэг. Бид энэхүү ажилд Марковицын сонгодог загварыг Паретогийн оновчлолтой холбон, багцын өгөөжийг нэмэгдүүлэх, эрсдлийг багасгахын тулд олон зорилтот оновчлолын бодлогыг авч үзлээ. Олон зорилтот оновчлолын Парето шийдүүд нь багцын хамгийн их өсөлтийг тодорхойлох асуудалд хариу өгдөг гэдгийг бид харууллаа. Тоон үр дүнг Матлаб ашиглан гаргасан. Түлхүүр үгс: Марковицийн онол, Олон-зорилтот оптимизаци &nbsp

Simulation on Sangaku problem using optimization methods

Sangaku problem is one of Japanese Temple Geometry problems which was studied in Hidetoshi Fukugawa[1]. One of the Sangaku problem is packing 6 equal circles in rectangle of 1:1.934798 size. We examine the problem from a view point of optimization theory and algorithm. We show that Sangaku optimization problem belongs to a class of nonconvex optimization and propose a penalty method for solving the problem numerically. In numerical expirements, we consider equal and unequal 6 circles. Computational results obtained on Python Jupyter Notebook are provided. Сангаку бодлогыг оптимизацийн аргаар бодох нь Хураангуй: Сангаку бодлого нь Японы эртний геометрийн бодлого юм. Энэхүү судалгаандаа бид Хидетоши Фукугава эрдэмтний судалсан 1:1.934798 хэмжээтэй тэгш өнцөгтөд 6 ижил тойрог багтаах сангаку бодлогыг авч үзэв. Энэ бодлогыг хучилтын бодлогын хүрээнд оптимизацийн аргаар бодсон ба энэ нь гүдгэр бус максимумчлалын бодлого болно.Бодлогыг өргөтгөж, ижил бус 6 тойргын хувьд бодож торгуулийн функцийн аргаар нэмж тооцооллыг хийв. Python Jupyter Notebook программ дээр тооцооллыг хийж үр дүнг гаргасан болно. Түлхүүр үгс: Сангаку бодлого, хучилтын бодлого, тойрог, оптимизацийн арг

A global optimization approach to fractional optimal control

In this paper, we consider a fractional optimal control problem governed by system of linear differential equations, where its cost function is expressed as the ratio of convex and concave functions. The problem is a hard nonconvex optimal control problem and application of Pontriyagin's principle does not always guarantee finding a global optimal control. Even this type of problems in a finite dimensional space is known as NP hard. This optimal control problem can, in principle, be solved by Dinkhelbach algorithm [10]. However, it leads to solving a sequence of hard D.C programming problems in its finite dimensional analogy. To overcome this difficulty, we introduce a reachable set for the linear system. In this way, the problem is reduced to a quasiconvex maximization problem in a finite dimensional space. Based on a global optimality condition, we propose an algorithm for solving this fractional optimal control problem and we show that the algorithm generates a sequence of local optimal controls with improved cost values. The proposed algorithm is then applied to several test problems, where the global optimal cost value is obtained for each case

AMO { AdvancedModeling and Optimization, Volume 6, Number 1, 2004

A method for synthesizing a string of letters in script style is developed based on the minimum principle for handwriting, whereby the pen motion is organized so as to minimize the integral of squared jerk subject to visiting a series of spatial points. Let the series of spatial points characterizing a string be a concatenation of those taken from individual letters. The pen motion is best expressed in the form of a linear combination of quintic B-splines. Then, synthesis of the pen motion to write the string is reduced to optimization of the times at whichthepenvisitsthecharacteristic points. The main contribution of this paper isanumber of recurrence formulae that makeitpossible to evaluate arithmetically the gradients of the integral of squared jerk with respect to visiting times. An optimization algorithm is compiled incorporating the recurrence formulae in a gradient method. Several examples of synthesized letter strings are presented

Joint Power Control, Base Station Assignment, and Channel Assignment in Cognitive Femtocell Networks

Cognitive radio and femtocells are recent technology breakthroughs that aim to achieve throughput improvement by means of spectrum management and interference mitigation, respectively. However, these technologies are limited by the former's susceptibility to interference and the latter's dependence on bandwidth availability. In this paper, we overcome these limitations by integrating cognitive radio and femtocell technology and exploring its feasibility and throughput improvement. To realize this, we propose an integrated architecture and formulate a multiobjective optimization problem with mixed integer variables for the joint power control, base station assignment, and channel assignment scheme. In order to find a pareto optimal solution, a weighted sum approach was used. Based on numerical results, the optimization framework is found to be both stable and converging. Simulation studies further show that the proposed architecture and optimization framework improve the aggregate throughput as the client population rises, hence confirming the successful and beneficial integration of these technologies.</p