Solving mixed-integer nonlinear programming problems using improved genetic algorithms

Tawan Wasanapradit,Thongchai Srinophakun,Nalinee Mukdasanit,Nachol Chaiyaratana 한국화학공학회 2011 Korean Journal of Chemical Engineering Vol.28 No.1

Abstract−This paper proposes a method for solving mixed-integer nonlinear programming problems to achieve or approach the optimal solution by using modified genetic algorithms. The representation scheme covers both integer and real variables for solving mixed-integer nonlinear programming, nonlinear programming, and nonlinear integer programming. The repairing strategy, a secant method incorporated with a bisection method, plays an important role in converting infeasible chromosomes to feasible chromosomes at the constraint boundary. To prevent premature convergence,the appropriate diversity of the structures in the population must be controlled. A cross-generational probabilistic survival selection method (CPSS) is modified for real number representation corresponding to the representation scheme. The efficiency of the proposed method was validated with several numerical test problems and showed good agreement.

A Novel Two-Step Tensegrity Topology-Finding Method Based on Mixed Integer Programming and Nonlinear Programming

Xian Xu,Shaoxiong Huang,Tingting Shu,Yafeng Wang,Yao-Zhi Luo 한국강구조학회 2022 International Journal of Steel Structures Vol.22 No.4

A two-step topology-fi nding method based on mixed integer programming and nonlinear programming is proposed for tensegrity structures. In the fi rst step, feasible and symmetric strut connectivities are obtained through a ground structure method combined with mixed integer programming; in the second step, the cable connectivities are optimized through nonlinear programming to obtain a feasible tensegrity structure. The same ground structure used in the fi rst step is adopted in the second step, which is benefi cial to fi nd a more symmetric cable layout. The independent continuous mapping method is used in the second step to convert the 0–1 binary variables of cable connectivities to continuous variables to transform the combinatorial optimization problem into a nonlinear programming problem. The number of strut lengths is adopted as a control parameter and a symmetry objective function is proposed to generate a variety of regular and symmetric tensegrity structures. A multi-stage computation scheme is proposed to improve the computational effi ciency. Typical examples are carried out to validate the proposed method. The computational effi ciency of the method is benchmarked with existing methods fully based on mixed integer programming. Results demonstrate that the computational effi ciency of the proposed method is signifi cantly improved compared to the existing methods.

A GENETIC ALGORITHM BASED ON OPTIMALITY CONDITIONS FOR NONLINEAR BILEVEL PROGRAMMING PROBLEMS

Hecheng Li,Yuping Wang 한국전산응용수학회 2010 Journal of applied mathematics & informatics Vol.28 No.3

For a class of nonlinear bilevel programming problems in which the follower's problem is linear, the paper develops a genetic algorithm based on the optimality conditions of linear programming. At first,we denote an individual by selecting a base of the follower's linear programming, and use the optimality conditions given in the simplex method to denote the follower's solution functions. Then, the follower's problem and variables are replaced by these optimality conditions and the solution functions, which makes the original bilevel programming become a single-level one only including the leader's variables. At last, the single-level problem is solved by using some classical optimization techniques, and its objective value is regarded as the fitness of the individual. The numerical results illustrate that the proposed algorithm is efficient and stable.

A GENETIC ALGORITHM BASED ON OPTIMALITY CONDITIONS FOR NONLINEAR BILEVEL PROGRAMMING PROBLEMS

Li, Hecheng,Wang, Yuping The Korean Society for Computational and Applied M 2010 Journal of applied mathematics & informatics Vol.28 No.3

For a class of nonlinear bilevel programming problems in which the follower's problem is linear, the paper develops a genetic algorithm based on the optimality conditions of linear programming. At first, we denote an individual by selecting a base of the follower's linear programming, and use the optimality conditions given in the simplex method to denote the follower's solution functions. Then, the follower's problem and variables are replaced by these optimality conditions and the solution functions, which makes the original bilevel programming become a single-level one only including the leader's variables. At last, the single-level problem is solved by using some classical optimization techniques, and its objective value is regarded as the fitness of the individual. The numerical results illustrate that the proposed algorithm is efficient and stable.

Inelastic vector finite element analysis of RC shells

Min, Chang-Shik,Gupta, Ajaya Kumar Techno-Press 1996 Structural Engineering and Mechanics, An Int'l Jou Vol.4 No.2

Vector algorithms and the relative importance of the four basic modules (computation of element stiffness matrices, assembly of the global stiffness matrix, solution of the system of linear simultaneous equations, and calculation of stresses and strains) of a finite element computer program for inelastic analysis of reinforced concrete shells are presented. Performance of the vector program is compared with a scalar program. For a cooling tower problem, the speedup factor from the scalar to the vector program is 34 for the element stiffness matrices calculation, 25.3 for the assembly of global stiffness matrix, 27.5 for the equation solver, and 37.8 for stresses, strains and nodal forces computations on a Gray Y-MP. The overall speedup factor is 30.9. When the equation solver alone is vectorized, which is computationally the most intensive part of a finite element program, a speedup factor of only 1.9 is achieved. When the rest of the program is also vectorized, a large additional speedup factor of 15.9 is attained. Therefore, it is very important that all the modules in a nonlinear program are vectorized to gain the full potential of the supercomputers. The vector finite element computer program for inelastic analysis of RC shells with layered elements developed in the present study enabled us to perform mesh convergence studies. The vector program can be used for studying the ultimate behavior of RC shells and used as a design tool.

시나리오 가중 방법론과 비선형 다단계Stochastic Programming을 사용한 수소공급망 네트워크 운용 프레임워크

장재욱,이현수 한국지능시스템학회 2024 한국지능시스템학회논문지 Vol.34 No.2

수소는 탄소, 미세먼지 오염 등 환경 문제 뿐만 아니라 화석 연료의 가격 변동 문제를 해결하기 위한 대체 에너지원으로 주목받고 있다. 이에 따라 많은 나라들이 수소의 생산, 저장그리고 운송과 관련된 Hydrogen Supply Chain Network (HSCN)을 생성했다. 생성된HSCN을 통해 각 나라의 지역적인 환경의 제약과 한정된 자원의 문제를 만족하며 비용 감소를 위한 다양한 연구가 진행되었다. 다양한 HSCN 모델 설계 연구에도 불구하고 설계된많은 HSCN은 실제 환경에서 발생가능한 불확실성을 간과하고 있다. 특히 HSCN에서 수소의 수요와 운송 capacity는 실제 환경에서 변동성이 큰 요소들이며 이러한 변동성을 무시하고 생성된 전략의 적용은 신뢰성의 문제와 오차로 인한 큰 대처 비용의 위험을 가진다. 따라서 본 연구에서는 단계적으로 발생하는 수소 수요와 운송 capacity의 불확실성을 고려한 의사 결정을 위해 multi-stage HSCN Stochastic Programming (SP)이 제시된다. 또한multi-stage SP에서 발생되는 결정 변수들 사이의 연쇄 법칙과 비선형 제약식을 만족하는해를 도출하기 위해 Weighted Scenario Method (WSM)가 제안되고 적용된다. 도출된 해의효과성은 불확실성을 고려하지 않은 결정론적 모델과의 비교 실험을 통해 입증된다. Hydrogen is gaining attention as an alternative energy source not only to addressenvironmental issues such as carbon and particulate pollution but also to tackle theproblem of price fluctuations in fossil fuels. In response to this, many countrieshave established Hydrogen Supply Chain Networks (HSCN) related to theproduction, storage, and transportation of hydrogen. Through the generated HSCNs,various studies have been conducted to satisfy regional environmental constraintsand limited resources, aiming at cost reduction. Despite numerous design studies onvarious HSCN models, many designed HSCNs overlook the uncertainties that canoccur in real-world environments. Particularly, hydrogen demand and transportationcapacity in HSCNs are highly variable factors in actual environments. Neglectingsuch variability and applying generated strategies may pose risks of reliabilityissues and significant coping costs due to errors. Therefore, this study proposes a multi-stage HSCN Stochastic Programming (SP)to consider the uncertainties arising in hydrogen demand and transportationcapacity for decision-making. Additionally, a Weighted Scenario Method (WSM) isintroduced and applied to derive solutions satisfying the chain rule and nonlinearconstraints among decision variables in the multi-stage SP. The effectiveness ofthe derived solutions is demonstrated through comparative experiments withdeterministic models that do not consider uncertainty.

AN APPROACH FOR SOLVING NONLINEAR PROGRAMMING PROBLEMS

Basirzadeh, H.,Kamyad, A.V.,Effati, S. 한국전산응용수학회 2002 The Korean journal of computational & applied math Vol.9 No.2

In this paper we use measure theory to solve a wide range of the nonlinear programming problems. First, we transform a nonlinear programming problem to a classical optimal control problem with no restriction on states and controls. The new problem is modified into one consisting of the minimization of a special linear functional over a set of Radon measures; then we obtain an optimal measure corresponding to functional problem which is then approximated by a finite combination of atomic measures and the problem converted approximately to a finite-dimensional linear programming. Then by the solution of the linear programming problem we obtain the approximate optimal control and then, by the solution of the latter problem we obtain an approximate solution for the original problem. Furthermore, we obtain the path from the initial point to the admissible solution.

Hybrid design method for air-core solenoid with axial homogeneity

Huang, Li,Lee, Sangjin,Choi, Sukjin The Korea Institute of Applied Superconductivity a 2016 한국초전도저온공학회논문지 Vol.18 No.1

In this paper, a hybrid method is proposed to design an air-core superconducting solenoid system for 6 T axial uniform magnetic field using Niobium Titanium (NbTi) superconducting wire. In order to minimize the volume of conductor, the hybrid optimization method including a linear programming and a nonlinear programming was adopted. The feasible space of solenoid is divided by several grids and the magnetic field at target point is approximated by the sum of magnetic field generated by an ideal current loop at the center of each grid. Using the linear programming, a global optimal current distribution in the feasible space can be indicated by non-zero current grids. Furthermore the clusters of the non-zero current grids also give the information of probable solenoids in the feasible space, such as the number, the shape, and so on. Applying these probable solenoids as the initial model, the final practical configuration of solenoids with integer layers can be obtained by the nonlinear programming. The design result illustrates the efficiency and the flexibility of the hybrid method. And this method can also be used for the magnet design which is required the high homogeneity within several ppm (parts per million).

A Switching Control Strategy for Nonlinear Systems Under Uncertainty

Yu Yang,Jong Min Lee 제어로봇시스템학회 2013 제어로봇시스템학회 국제학술대회 논문집 Vol.2013 No.10

Nonlinear systems under uncertainty are difficult to regulate with guaranteed stability and optimality. This study presents a switching control strategy, which consists of robust control Lyapunov function-based predictive controller and approximate dynamic programming-based controller. The former guarantees the robust stability within a level set, referred to as region of attraction (ROA). The latter improves optimality and reduces computational complexity in solving Bellman equation when the system is outside the ROA. The suggested approach is illustrated on a continuous stirred tank reactor example.

A STACKELBERG MODEL FOR SERVER-PROXIES-USERS SYSTEMS

HAW, HAl SHAN,XIA, ZUN-QUAN 한국전산응용수학회 2005 Journal of applied mathematics & informatics Vol.17 No.1

A Server-Proxies-Users communication system is studied by using Stackelberg strategy theory of game. A new model, in which the server, proxies and users are not equal is established, and that is a three-level programming. The solution existence of the model is proved.