DOCSLIB.ORG

Particle Swarm Optimization for Constrained and Multiobjective Problems: a Brief Review

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Particle Swarm Optimization for Constrained and Multiobjective Problems: a Brief Review 2011 International Conference on Management and Artificial Intelligence IPEDR vol.6 (2011) © (2011) IACSIT Press, Bali, Indonesia Particle Swarm Optimization for Constrained and Multiobjective Problems: A Brief Review Nor Azlina Ab. Aziz Mohamad Yusoff Alias Faculty of Engineering and Technology Faculty of Engineering Multimedia University Multimedia University Malaysia Malaysia [email protected] [email protected] Ammar W. Mohemmed Kamarulzaman Ab. Aziz Knowledge Engineering & Discovery Research Institute Faculty of Management Auckland University of Technology Multimedia University New Zealand Malaysia [email protected] [email protected] Abstract— Particle swarm optimization (PSO) is an of PSO. Particles neighbourhood in PSO had been studied optimization method that belongs to the swarm intelligence from two perspectives; global neighbourhood (gBest) and family. It was initially introduced as continuous problem local neighbourhood (lBest). In gBest the particles are fully optimization tool. It has evolved to being applied to more connected therefore the particles search is directed by the complex multiobjective and constrained problem. This paper best particle of the swarm. While in lBest the particles are presents a systematic literature review of PSO for constrained connected to their neighbours only and their search is and multiobjective optimization problems. conducted by referring to the neighbourhood best. A particle in PSO has a position (X ) and velocity (V ). The position Keywords- Constrained optimization problems; multiobjective i i represents a solution suggested by the particle while velocity optimization problems; particle swarm optimization is the rate of changes of the next position with respect to I. INTRODUCTION current position. Initially these two values (position and velocity) are randomly initialised. In the subsequent One of successful optimization algorithms is particle iterations the search process is conducted by updating these swarm optimization (PSO). PSO main attractive feature is its values using the following equations: simple and straightforward implementation. PSO has been 1 2 2 = × 1 −××+−××+ (1) applied in multiple fields such as human tremor analysis for i w VV i randc XP ii )(() randc XP ig )(() biomedical engineering, electric power and voltage += (2) management and machine scheduling [1]. XX ii Vi The original PSO is proposed for optimization of single where i is particle’s number (i = 1,..,N; N: number of objective continuous problem. However the concept of PSO particles in the swarm). has been expanded to allow it to handle other optimization As what can be observed in Eq.1 velocity is influenced problems such as; binary, discrete, combinatorial, by; Pi – the best position found so far by the particle and Pg – constrained and multiobjective optimization. This paper will the best position found by the neighbouring particles. Pg review some of the works conducted in constrained and could be either the position of gBest or lBest. The Vi value is multiobjective PSO in section 4 and 5. However before that clamped to ±Vmax to prevent explosion. If the value of Vmax is the concept of PSO is discussed in the next section followed too large the exploration range is too wide however if it is by the concept of multiobjective and constrained problems. too small particles will favour local search [3]. c1 and c2 are The paper is then concluded in section 6. the learning factors to control the effect of the “best” factors of particles; Pi and Pg, typically both are set to 2 [1]. rand1() II. PARTICLE SWARM OPTIMIZATION and rand2() are two independent random numbers in the In 1995, James Kennedy and Russell Eberhart introduced range of [0.0,1.0]. The randomness provides energy to the particle swarm optimization (PSO). It is a swarm based particles. w is known as inertia weight, a term added to algorithm that mimics the social behaviour of organisms like improve PSO’s performance. The inertia weight controls birds and fishes. The success of an individual in these particles momentum so that, they can avoid continuing to communities is affected not only by its own effort but also explore the wide search space and switch to fine tuning when by the information shared by its surrounding neighbours. a good area is found [4]. This nature of the social behaviour is imitated by PSO using The particle position is updated using Eq.2 where the swarm of agents called particles [2]. The interaction of the velocity is added to the previous position. This shows how a particles with their neighbours is the key to the effectiveness 146 particle next search is launched from its previous position hk(X) are inequality and equality constraints enforced on the and the new search is influenced by its pass search [5]. problem, respectively. X is the search variables vector that The quality of the solution is evaluated by a fitness is bounded in the search space within the xmin and xmax values. function, which is a problem-dependent function. If the A constrained optimization could have only one objective, current solution is better than the fitness of Pi or Pg or both, meanwhile multiobjective optimization could be limited by the best value will be replaced by current solution its search space only. accordingly. This update process continues until stopping criterion is met, usually when either maximum iteration is IV. PSO FOR CONSTRAINED OPTIMIZATION PROBLEMS achieved or target solution is attained. When the stopping In their work, [6] categorised the evolutionary algorithm criterion is satisfied, the best particle found so far is taken as optimization methods for constrained problem into four the optimal solution (near optimal). The PSO algorithm is types: presented in Fig. 1. Initialize particles population; A) Preserve feasibility of solutions Do{ According to this approach only feasible solutions are Calculate fitness values of each particles using fitness function; generated to maintain feasibility. The search process is also Update pid if the current fitness value is better than pid; Determine pgd : choose the particle position with the best fitness restricted within only the feasible area. value of all the neighbors as the pgd; For each particle { B) Penalty functions Calculate particle velocity according to (1); In the penalty functions method, the constrained Update particle position according to (2); optimization problem is solved using unconstrained } } While maximum iteration or ideal fitness is not attained; optimization method by incorporating the constraints into the objective function thus transforming it into an unconstrained Fig. 1: PSO Algorithm problem. III. MULTIOBJECTIVE AND CONSTRAINED OPTIMIZATION C) Differentiate the feasible and infeasible solutions PROBLEMS Among common methods belonging to this group are Multiobjective optimization is a problem with many repairing infeasible solutions so that a feasible solution is objectives to be fulfilled and most of the time these produced and preferring feasible solutions over infeasible objectives are in conflict with each other. Whereas solutions. constrained optimization is an optimization problem with D) Hybrid methods one or more constraints to be obeyed. These types of problems are commonly faced in A hybrid method is a combination of at least two of the everyday life, for example in this situation: three methods described above. Mr. ABC wants to buy a new computer. He wants a Among the early research works on PSO for constrained computer with the best specifications but with the least optimization problem is by [7] where the simple idea of feasible solution preservation is used. There are two points cost. that differentiate this work from the basic PSO. The first one The problem faced by Mr. ABC is a multiobjective is that the particles are initialised with only feasible solutions. problem with two objectives; the first objective is to get a This step speeds up the search process. The second point is computer with the best specifications while the second the update of the best solutions (gBest and pBest). Only objective is to spend the least amount of money. Following is feasible solutions will be selected as the best values. This a similar problem in constrained optimization problem form: approach is simple and able to handle a wide variety of Ms. XYZ also wants to buy a computer, but she has a constrained problems. However, for problems with a small limited budget. She is aiming to get the best computer feasible region the initialisation process tends to consume a possible within her budget. lot of time thus delaying the search [8]. Ms. XYZ is facing a constrained optimization problem Penalty function technique is usually used in where she needs to buy a good computer but subject to a evolutionary algorithm for constrained optimization limited budget. problems [9]. The penalty function combines the constraints Mathematically these optimization problems can be with the objective function by adding penalty value to presented as follows: infeasible solutions (in minimization problems). In [10] the optimize :()X i= 1,..., l f i penalty functions methods are grouped into three classes: subject to : (X )≤ 0j = 1,..., p (3) c j (a) Static penalty functions (X )= 0 k = p +1,..., q This is the simplest approach for penalty functions. It hk gives fixed penalty value once a constraint is violated no T X = X ∈[,] matter how large or small the violations are. A better method (x1 , x 2 ,..., xn ) xmin xmax is to penalise the constraints violation based on how far the solution is from the feasible region (distance based penalty). f (X) are the l objectives to be optimized while c (X) and i j This can be done by multiplying the fixed penalty 147 coefficients with their constraint functions or by dividing the the constrained optimization problem into min-max problem violation into several levels with each level having its own by transforming the primal problem into its dual problem, penalty coefficient. e.g., for a minimization primal problem, the dual problem is maximization.
Load more

Recommended publications
  • Lecture 3 1 Geometry of Linear Programs
    ORIE 6300 Mathematical Programming I September 2, 2014 Lecture 3 Lecturer: David P. Williamson Scribe: Divya Singhvi Last time we discussed how to take dual of an LP in two different ways. Today we will talk about the geometry of linear programs. 1 Geometry of Linear Programs First we need some definitions. Definition 1 A set S ⊆ <n is convex if 8x; y 2 S, λx + (1 − λ)y 2 S, 8λ 2 [0; 1]. Figure 1: Examples of convex and non convex sets Given a set of inequalities we define the feasible region as P = fx 2 <n : Ax ≤ bg. We say that P is a polyhedron. Which points on this figure can have the optimal value? Our intuition from last time is that Figure 2: Example of a polyhedron. \Circled" corners are feasible and \squared" are non feasible optimal solutions to linear programming problems occur at \corners" of the feasible region. What we'd like to do now is to consider formal definitions of the \corners" of the feasible region. 3-1 One idea is that a point in the polyhedron is a corner if there is some objective function that is minimized there uniquely. Definition 2 x 2 P is a vertex of P if 9c 2 <n with cT x < cT y; 8y 6= x; y 2 P . Another idea is that a point x 2 P is a corner if there are no small perturbations of x that are in P . Definition 3 Let P be a convex set in <n. Then x 2 P is an extreme point of P if x cannot be written as λy + (1 − λ)z for y; z 2 P , y; z 6= x, 0 ≤ λ ≤ 1.
    [Show full text]
  • Extreme Points and Basic Solutions
    EXTREME POINTS AND BASIC SOLUTIONS: In Linear Programming, the feasible region in Rn is defined by P := {x ∈ Rn | Ax = b, x ≥ 0}. The set P , as we have seen, is a convex subset of Rn. It is called a convex polytope. The term convex polyhedron refers to convex polytope which is bounded. Polytopes in two dimensions are often called polygons. Recall that the vertices of a convex polytope are what we called extreme points of that set. Recall that extreme points of a convex set are those which cannot be represented as a proper convex combination of two other (distinct) points of the convex set. It may, or may not be the case that a convex set has any extreme points as shown by the example in R2 of the strip S := {(x, y) ∈ R2 | 0 ≤ x ≤ 1, y ∈ R}. On the other hand, the square defined by the inequalities |x| ≤ 1, |y| ≤ 1 has exactly four extreme points, while the unit disk described by the ineqality x2 + y2 ≤ 1 has infinitely many. These examples raise the question of finding conditions under which a convex set has extreme points. The answer in general vector spaces is answered by one of the “big theorems” called the Krein-Milman Theorem. However, as we will see presently, our study of the linear programming problem actually answers this question for convex polytopes without needing to call on that major result. The algebraic characterization of the vertices of the feasible polytope confirms the obser- vation that we made by following the steps of the Simplex Algorithm in our introductory example.
    [Show full text]
  • The Feasible Region for Consecutive Patterns of Permutations Is a Cycle Polytope
    Séminaire Lotharingien de Combinatoire XX (2020) Proceedings of the 32nd Conference on Formal Power Article #YY, 12 pp. Series and Algebraic Combinatorics (Ramat Gan) The feasible region for consecutive patterns of permutations is a cycle polytope Jacopo Borga∗1, and Raul Penaguiaoy 1 1Department of Mathematics, University of Zurich, Switzerland Abstract. We study proportions of consecutive occurrences of permutations of a given size. Specifically, the feasible limits of such proportions on large permutations form a region, called feasible region. We show that this feasible region is a polytope, more precisely the cycle polytope of a specific graph called overlap graph. This allows us to compute the dimension, vertices and faces of the polytope. Finally, we prove that the limits of classical occurrences and consecutive occurrences are independent, in some sense made precise in the extended abstract. As a consequence, the scaling limit of a sequence of permutations induces no constraints on the local limit and vice versa. Keywords: permutation patterns, cycle polytopes, overlap graphs. This is a shorter version of the preprint [11] that is currently submitted to a journal. Many proofs and details omitted here can be found in [11]. 1 Introduction 1.1 Motivations Despite not presenting any probabilistic result here, we give some motivations that come from the study of random permutations. This is a classical topic at the interface of combinatorics and discrete probability theory. There are two main approaches to it: the first concerns the study of statistics on permutations, and the second, more recent, looks arXiv:2003.12661v2 [math.CO] 30 Jun 2020 for the limits of permutations themselves.
    [Show full text]
  • Linear Programming
    Stanford University | CS261: Optimization Handout 5 Luca Trevisan January 18, 2011 Lecture 5 In which we introduce linear programming. 1 Linear Programming A linear program is an optimization problem in which we have a collection of variables, which can take real values, and we want to find an assignment of values to the variables that satisfies a given collection of linear inequalities and that maximizes or minimizes a given linear function. (The term programming in linear programming, is not used as in computer program- ming, but as in, e.g., tv programming, to mean planning.) For example, the following is a linear program. maximize x1 + x2 subject to x + 2x ≤ 1 1 2 (1) 2x1 + x2 ≤ 1 x1 ≥ 0 x2 ≥ 0 The linear function that we want to optimize (x1 + x2 in the above example) is called the objective function.A feasible solution is an assignment of values to the variables that satisfies the inequalities. The value that the objective function gives 1 to an assignment is called the cost of the assignment. For example, x1 := 3 and 1 2 x2 := 3 is a feasible solution, of cost 3 . Note that if x1; x2 are values that satisfy the inequalities, then, by summing the first two inequalities, we see that 3x1 + 3x2 ≤ 2 that is, 1 2 x + x ≤ 1 2 3 2 1 1 and so no feasible solution has cost higher than 3 , so the solution x1 := 3 , x2 := 3 is optimal. As we will see in the next lecture, this trick of summing inequalities to verify the optimality of a solution is part of the very general theory of duality of linear programming.
    [Show full text]
  • Math 112 Review for Exam Ii (Ws 12 -18)
    MATH 112 REVIEW FOR EXAM II (WS 12 -18) I. Derivative Rules • There will be a page or so of derivatives on the exam. You should know how to apply all the derivative rules. (WS 12 and 13) II. Functions of One Variable • Be able to find local optima, and to distinguish between local and global optima. • Be able to find the global maximum and minimum of a function y = f(x) on the interval from x = a to x = b, using the fact that optima may only occur where f(x) has a horizontal tangent line and at the endpoints of the interval. Step 1: Compute the derivative f’(x). Step 2: Find all critical points (values of x at which f’(x) = 0.) Step 3: Plug all the values of x from Step 2 that are in the interval from a to b and the endpoints of the interval into the function f(x). Step 4: Sketch a rough graph of f(x) and pick off the global max and min. • Understand the following application: Maximizing TR(q) starting with a demand curve. (WS 15) • Understand how to use the Second Derivative Test. (WS 16) If a is a critical point for f(x) (that is, f’(a) = 0), and the second derivative is: f ’’(a) > 0, then f(x) has a local min at x = a. f ’’(a) < 0, then f(x) has a local max at x = a. f ’’(a) = 0, then the test tells you nothing. IMPORTANT! For the Second Derivative Test to work, you must have f’(a) = 0 to start with! For example, if f ’’(a) > 0 but f ’(a) ≠ 0, then the graph of f(x) is concave up at x = a but f(x) does not have a local min there.
    [Show full text]
  • The Feasible Region for Consecutive Patterns of Permutations Is a Cycle Polytope
    Algebraic Combinatorics Draft The feasible region for consecutive patterns of permutations is a cycle polytope Jacopo Borga & Raul Penaguiao Abstract. We study proportions of consecutive occurrences of permutations of a given size. Specifically, the limit of such proportions on large permutations forms a region, called feasible region. We show that this feasible region is a polytope, more precisely the cycle polytope of a specific graph called overlap graph. This allows us to compute the dimension, vertices and faces of the polytope, and to determine the equations that define it. Finally we prove that the limit of classical occurrences and consecutive occurrences are in some sense independent. As a consequence, the scaling limit of a sequence of permutations induces no constraints on the local limit and vice versa. (1,0,0,0,0,0) 1 1 (0; 2 ; 2 ; 0; 0; 0) 1 1 1 1 (0; 0; 2 ; 2 ; 0; 0) (0; 2 ; 0; 2 ; 0; 0) 1 1 (0; 0; 0; 2 ; 2 ; 0) arXiv:1910.02233v2 [math.CO] 30 Jun 2020 Figure 1. The four-dimensional polytope P3 given by the six pat- terns of size three (see Eq. (2) for a precise definition). We highlight in light-blue one of the six three-dimensional facets of P3. This facet is a pyramid with square base. The polytope itself is a four-dimensional pyramid, whose base is the highlighted facet. This paper has been prepared using ALCO author class on 1st July 2020. Keywords. Permutation patterns, cycle polytopes, overlap graphs. Acknowledgements. This work was completed with the support of the SNF grants number 200021- 172536 and 200020-172515.
    [Show full text]
  • Mathematical Modelling and Applications of Particle Swarm Optimization
    Master’s Thesis Mathematical Modelling and Simulation Thesis no: 2010:8 Mathematical Modelling and Applications of Particle Swarm Optimization by Satyobroto Talukder Submitted to the School of Engineering at Blekinge Institute of Technology In partial fulfillment of the requirements for the degree of Master of Science February 2011 Contact Information: Author: Satyobroto Talukder E-mail: [email protected] University advisor: Prof. Elisabeth Rakus-Andersson Department of Mathematics and Science, BTH E-mail: [email protected] Phone: +46455385408 Co-supervisor: Efraim Laksman, BTH E-mail: [email protected] Phone: +46455385684 School of Engineering Internet : www.bth.se/com Blekinge Institute of Technology Phone : +46 455 38 50 00 SE – 371 79 Karlskrona Fax : +46 455 38 50 57 Sweden ii ABSTRACT Optimization is a mathematical technique that concerns the finding of maxima or minima of functions in some feasible region. There is no business or industry which is not involved in solving optimization problems. A variety of optimization techniques compete for the best solution. Particle Swarm Optimization (PSO) is a relatively new, modern, and powerful method of optimization that has been empirically shown to perform well on many of these optimization problems. It is widely used to find the global optimum solution in a complex search space. This thesis aims at providing a review and discussion of the most established results on PSO algorithm as well as exposing the most active research topics that can give initiative for future work and help the practitioner improve better result with little effort. This paper introduces a theoretical idea and detailed explanation of the PSO algorithm, the advantages and disadvantages, the effects and judicious selection of the various parameters.
    [Show full text]
  • Lesson . Geometry and Algebra of “Corner Points”
    SAh§þ Linear Programming Spring z§zË Uhan Lesson Ëþ.GeomeTry and Algebra of “Corner Points” §W arm up Example Ë. Consider The sysTem of equaTions hxË + xz − 6xh = Ë6 xË + þxz = Ë (∗) −zxË + ËËxh = −z ⎛ h Ë −6⎞ LeT A = ⎜ Ë þ § ⎟. We have ThaT deT(A) = Ç . ⎝−z § ËË ⎠ ● Does (∗) have a unique soluTion, no soluTions, or an inûniTe number of soluTions? ● Are The row vecTors of A linearly independent? How abouT The column vecTors of A? ● WhaT is The rank of A? Does A have full row rank? ËO verview xz ● Due To convexiTy, local opTimal soluTions of LPs are global opTimal soluTions þ ⇒ Improving search ûnds global opTimal soluTions of LPs ● _e simplex meThod: improving search among “corner points” of The h ↑ (Ë) feasible region of an LP z (z ) ↓ ● How can we describe “corner points” of The feasible region of an LP? Ë ● For LPs, is There always an opTimal soluTion ThaT is a “corner poinT”? xË Ë z h Ë z Polyhedra and exTreme points ● A polyhedron is a seT of vecTors x ThaT saTisfy a ûniTe collecTion of linear consTraints (equaliTies and inequaliTies) ○ Also referred To as a polyhedral seT ● In parTicular: ● Recall: The feasible region of an LP – a polyhedron – is a convex feasible region ● Given a convex feasible region S, a soluTion x ∈ S is an exTreme poinT if There does noT exisT Two disTincT soluTions y, z ∈ S such ThaT x is on The line segmenT joining y and z ○ i.e. There does noT exisT λ ∈ (§, Ë) such ThaT x = λy + (Ë − λ)z Example z.
    [Show full text]
  • Geometry and Visualizations of Linear Programs (PDF)
    15.053/8 February 12, 2013 Geometry and visualizations of linear programs © Wikipedia User: 4C. License CC BY-SA. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/. 1 Quotes of the day “You don't understand anything until you learn it more than one way.” Marvin Minsky “One finds limits by pushing them.” Herbert Simon 2 Overview Views of linear programming – Geometry/Visualization – Algebra – Economic interpretations 3 What does the feasible region of an LP look like? Three 2-dimensional examples 4 Some 3-dimensional LPs Courtesy of Wolfram Research, Inc. Used with permission. Source: Weisstein, Eric W. "Convex Polyhedron." From MathWorld -- A Wolfram Web Resource. 5 Goal of this Lecture: visualizing LPs in 2 and 3 dimensions. What properties does the feasible region have? – convexity – corner points What properties does an optimal solution have? How can one find the optimal solution: – the “geometric method” – The simplex method Introduction to sensitivity analysis – What happens if the RHS changes? 6 A Two Variable Linear Program (a variant of the DTC example) objective z = 3x + 5y 2x + 3y 10 (1) Constraints x + 2y 6 (2) x + y 5 (3) x 4 (4) y 3 (5) x, y 0 (6) 7 Finding an optimal solution Introduce yourself to your partner Try to find an optimal solution to the linear program, without looking ahead. 8 Inequalities A single linear inequality determines a unique y half-plane 5 4 x + 2y 6 3 2 1 x 1 2 3 4 5 6 9 Graphing the Feasible Region y Graph the Constraints: 2x+ 3y 10 (1) 5 x 0 , y 0.
    [Show full text]
  • B1 Optimization – Solutions
    B1 Optimization – Solutions A. Zisserman, Michaelmas Term 2018 1. The Rosenbrock function is f(x; y) = 100(y − x2)2 + (1 − x)2 (a) Compute the gradient and Hessian of f(x; y). (b) Show that that f(x; y) has zero gradient at the point (1; 1). (c) By considering the Hessian matrix at (x; y) = (1; 1), show that this point is a mini- mum. (a) Gradient and Hessian 400x3 − 400xy + 2x − 2 1200x2 − 400y + 2 −400x rf = H = 200(y − x2) −400x 200 (b) gradient at the point (1; 1) 400x3 − 400xy + 2x − 2 0 rf = = 200(y − x2) 0 (b) Hessian at the point (1; 1) 1200x2 − 400y + 2 −400x 802 −400 H = = −400x 200 −400 200 Examine eigenvalues: • det is positive, so eigenvalues have same sign (thus not saddle point) • trace is positive, so eigenvalues are positive • Thus a minimum • λ1 = 1001:6006; λ2 = 0:39936077 1 2. In Newton type minimization schemes the update step is of the form δx = −H−1g where g = rf. By considering g.δx compare convergence of: (a) Newton, to (b) Gauss Newton for a general function f(x) (i.e. where H may not be positive definite). A note on positive definite matrices An n × n symmetric matrix M is positive definite if • x>Mx > 0 for all non-zero vectors x • All the eigen-values of M are positive In each case consider df = g:δx. This should be negative for convergence. (a) Newton g:δx = −g>H−1g Can be positive if H not positive definite.
    [Show full text]
  • Load Feasible Region Determination by Using Adaptive Particle Swarm Optimization
    Article Load Feasible Region Determination by Using Adaptive Particle Swarm Optimization Patchrapa Wongchaia and Sotdhipong Phichaisawatb,* Department of Electrical Engineering, Faculty of Engineering, Chulalongkorn University, Bangkok 10330, Thailand E-mail: [email protected], [email protected] Abstract. The proposed method determines points in a feasible region by using an adaptive particle swarm optimization in order to solve the boundary region which represented by the obtained points. This method is also used for calculating a large-scale power system. In any contingency case, it will be illustrated with an x-axis and y-axis space which is given by the power flow analysis. In addition, this presented approach in this paper not only demonstrates the optimal points through the boundary tracing method of the feasible region but also presents the boundary points obtained the particle swarm optimization. Moreover, decreasing loss function and operational physical constraints such as voltage level, equipment specification are all simultaneously considered. The points in the feasible region are also determined the boundary points which a point happening a contingency in the power system is already taken into account and the stability of load demand is ascertained into the normal operation, i.e. the power system can be run without violation. These feasible points regulate the actions of the system and the robustness of the operating points. Finally, the proposed method is evaluated on the test system to examine the impact of system parameters relevant to generation and consumption. Keywords: Particle swarm optimization, feasible region, boundary tracing method, power flow solution space. ENGINEERING JOURNAL Volume 23 Issue 6 Received 28 July 2019 Accepted 30 September 2019 Published 30 November 2019 Online at http://www.engj.org/ DOI:10.4186/ej.2019.23.6.239 DOI:10.4186/ej.2019.23.6.239 1.
    [Show full text]
  • Modeling with Nonlinear Programming
    CHAPTER 4 Modeling with Nonlinear Programming By nonlinear programming we intend the solution of the general class of problems that can be formulated as min f(x) subject to the inequality constraints g (x) 0 i ≤ for i = 1,...,p and the equality constraints hi(x) = 0 for i = 1,...,q. We consider here methods that search for the solution using gradient information, i.e., we assume that the function f is differentiable. EXAMPLE 4.1 Given a fixed area of cardboard A construct a box of maximum volume. The nonlinear program for this is min xyz subject to 2xy + 2xz + 2yz = A EXAMPLE 4.2 Consider the problem of determining locations for two new high schools in a set of P subdivisions Nj . Let w1j be the number of students going to school A and w2j be the number of students going to school B from subdivision Nj. Assume that the student capacity of school A is c1 and the capacity of school B is c2 and that the total number of students in each subdivision is rj . We would like to minimize the total distance traveled by all the students given that they may attend either school A or B. It is possible to construct a nonlinear program to determine the locations (a, b) and (c, d) of high schools A and B, respectively assuming the location of each subdivision Ni is modeled as a single point denoted (xi,yi). 1 1 P 2 2 min w (a x )2 + (b y )2 + w (c x )2 + (d y )2 1j − j − j 2j − j − j Xj=1 64 Section 4.1 Unconstrained Optimization in One Dimension 65 subject to the constraints w c ij ≤ i Xj w1j + w2j = rj for j = 1,...,P .
    [Show full text]