Generalized Linear Fractional Programming – Theory, Algorithms and Applications (I)

九十四年度工業工程與管理學門專題研究計畫成果報告九十四年度工業工程與管理學門專題研究計畫成果報告 Generalized Linear Fractional Programming –Theory, Algorithms and Applications (I) 廣義型線性分式規劃– 理論、演算法與應用(I) 執行單位：國立聯合大學經營管理學系計畫主持人：吳光耀計畫編號：NSC94-2213-E-239-005 日期: 2006/11/11

. Preface . . . • Generalized Fractional Programming (GFP) • Minimize the largest or maximize the smallest multi-ratio functions • Consider a problem with l fractional functions, u constraints and v variables: • Y, Gk > 0 • have been studied extensively (Frenk and Schaible, 2001) • Generalized Linear Fractional Programming (GLFP) • Fk , Gk and Hi are affine functions

Conti. Y (Gray plane) (y) surf The minimal solution y2 y1 Preface . . . The minimal solution is a vertex lying on the surface of the epigraph. • An example of GLFP min (y1, y2, y3)=max{ , , } s.t. 3 y1 + 4 y2 + y3 12 = 0, y10, y20, y30.

Agenda • Introduction • Theory, Algorithms and Applications • Objective • Methodology • Problem Statement • Basic Solutions and Optimality • Pivoting for an Improvement • Optimality Condition • Dual Approach • Results and Discussions • Numerical Results • Discussions and Summary • Conclusion

1. Introduction Applications • A survey (Crouzeix and Ferland, 1991) • The Von Neumann’s model of an expending economy • The best rational approximation • Goal programming and multi-criteria optimization • The others • Polynomial max-min (Tigan el at., 2001) • Fuzzy linear optimization (Wang and Wu, 2004) • In this 3-year plan • Financial planning • Throughput optimization • The efficiency of the employed algorithms is critical in applications.

1. Introduction Theory • Duality • Farkas’ Lemma (Jagannathan and Schaible, 1983)  Alternative theorems • Quasiconvexity (Crouzeix, et al., 1983)  Generalized convexity and generalized monotonicity • Reveal a lower bound • Optimality • A local minimum is global (Crouzeix, et al., 1983)  is strictly quasiconvex • An equivalent parametric program (Crouzeix, et al., 1983) Generalizations of Dinkelbach (1967) • Find the root of the parametric problem

1. Introduction Algorithms • The standard method • Dinkelbach-type approach (Crouzeix et al,1985) • Its hybridization approach (Ferland and Potvin, 1985) • Its dual approach (Barros et al, 1996) • The others • Interval-type algorithms (Bernard and Ferland,1989) • Interior-point approach (Freund and Jarre, 1995) • Unified monotonic approach (Phuong and Tuy, 2003) • In this study • Basis-based approach

1. Introduction Remark on Dinkelbach-type Approach a non-linear program a linear program (LP) There are many algorithms available for GLFP problems and the overwhelming majority were developed for problems of GFP. GFP GLFP  Problems Sub-problems  Solution Approaches • A parameterized sub-problem is required for each iteration.

1. Introduction Objective in the first year • Develop a basis-based approach with extension to • Dual model • Sensitivity analysis • Linearly constrained optimization A vertex … Parametric basic solutions LP0  Our basis-based approach  Dinkelbach-type approach LP1 LP2 … Parameterized sub-problems

2. Methodology Problem Statement • Augmented model inf { :A()x=d(), x0} A() =   d() =   xR n m = l+u, n = l+v

2. Methodology Basic Solutions and Optimality A basic solution is corresponding to a vertex of the epigraph. Assume the vertices are non-degenerate. • A basic solution (x,) • nm+1 zero-valued x-variables called nonbasic variables Let Z={z1,z2, …,znm+1} denote the set of the indices of the nonbasic. • The other m1 nonzero x-variables are basic variables. Let B={b1,b2,…,bm1} denote the set of the indices of the basic. • If xB>0, it is a basic feasible solution. • Theorem 1: The optimal value is obtained by a basic feasible solution.

2. Methodology Pivoting for an Improvement A basic solution (x,) with (Z,B) • Correspond to an mm nonlinear system AB()x=d(), x0 • Introduce a non-basic variables into the system • Choose an x for Z as the entering basic variable for pivoting i.e. consider nonbasic x to be increased from zero to positive • Arrange two index sets of P=B{} and R=Z\{}, in which =p • Identify the solution as a BF (x,) with (R,P,,) Z={ z1 z2 . . . … zn-m+1} B={ b1 b2 . . . bm-1 } R={ r1 r2 . . . rn-m} P={ p1 p2 . . . p … pm }

2. Methodology Optimality Condition • Now xp()=0 and xP()=AP()1 ([A1]PxP()+d1) • If  xp()>0, the nonbasic variable cannot be increased from zero. • Theorem 2:The necessary and sufficient condition for (x,) of (R,P,,) to be optimal is that xp() > 0 and q  0, whereQ=AP()1 AR(). • Develop a solution procedure for exploring basic feasible solutions in an iterative matter until an optimal point is reached (Wu and Wang, 2006)

2. Methodology Dual Approach • Via alternative theorems, the dual problem sup{: = 0, 0, 0, 0, 0, R } • Lemma 1. Suppose that [yT sT] and [T T T ] are optimal for the primal and the dual, respectively. Then it holds that yii=0 for all iV and sii=0 for all iL. • Define basic solutions in the dual model • Develop a basis-based algorithm for the dual (Wu, 2006b)

3. Results and Discussions Numerical Results • The proposed basis-based approach to the primal problem • Employ three Dinkelbach-type algorithms (sub-problem-based approach) • Compared with 60 problems • Algorithm BASIS significantly outperforms the others

3. Results and Discussions Other Results with Discussions < Basis-based approach to the primal problem> (Wu and Wang, 2006, JOTA) • Outperform the Dinkelbach-type algorithms • Various methods tantamount to gradient search for refining • By resolving duality in a basis, an eigenvalue method has been proposed (Wu, 2006c, ORSTW). Solve large scale problems in a robust way • Be similar to the conventional simplex method • Extension to some problems with linear constraints • A unified simplex-type approach has been established (Wu, 2005, CIIE; Wu, 2006a, JCIIE). Prevent cycling in solving linear programs

3. Results and Discussions Other Results with Discussions < Basis-based approach to the primal problem > • Facilitate the development of sensitivity analysis • Applications to decision-making in some practical problems • An application to throughput planning in a production system has been demonstrated (Wu and Lai, 2006, CCIE). Develop a GLFP-aided interactive system < Basis-based approach to the dual problem> (Wu, 2006b, INFORMS) • Be adaptive to solve the fractional problems with a considerable number of linear ratios • Numerical results in large scale problems remain to be done.

Theory, Algorithms and Applications Generalized Linear Fractional Programming Conclusion • The first-year Theory: Basic solutions, duality and optimality Algorithms: Basis-based approach to • Primal problem (Wu and Wang, 2006, JOTA) • Dual problem (Wu, 2006b, INFORMS) • Dual basis (Wu, 2006c, TWSOR) Applications: • Optimization with linear constraints (Wu, 2005, CIIE; Wu, 2006a, JCIIE) • Throughput planning problem (Wu and Lai, 2006, CCIE) • Future study • Theory: Generalized convexity • Algorithms: Numerical experiment • Applications: Financial planning, Throughput optimization

Generalized Linear Fractional Programming – Theory, Algorithms and Applications (I)