Efficient Algorithms for Solving Nonlinear Fractional Programming
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A New Algorithm for Generalized Fractional Programs 1
Mathematical Programming 72 (1996) 147-175 A new algorithm for generalized fractional programs 1 A.I. Barros a, J.B.G. Frenk ~'* , S. Schaible c, S. Zhang b a D.E.I.O.. FacuMade de CiOncias, Uniuersi~hule de Lisboa. Campo Grande. 1700 Lisboa. Portugal b Econometrtc.... lnstttute. Erasmus Umuerstty. P.O. Box 1738. 3000 DR Rotterdam. The Netherlands c University of Cal([ornia at Riverside. Riverside, CA 92521, USA Received 2 February 1994 Abstract A new dual problem for convex generalized fractional programs with no duality gap is presented and it is shown how this dual problem can be efficiently solved using a parametric approach. The resulting algorithm can be seen as "'dual'" to the Dinkelbach-type algorithm for generalized fractional programs since it approximates the optimal objective value of the dual (primal) problem from below. Convergence results for this algorithm are derived and an easy condition to achieve superlinear convergence is also established. Moreover, under some additional assumptions the algorithm also recovers at the same time an optimal solution of the primal problem. We also consider a variant of this new algorithm, based on scaling the "'dual" parametric function, The numerical results, in case of quadratic-linear ratios and linear constraints, show that the performance of the new algorithm and its scaled version is superior to that of the Dinkelbach-type algorithms. From the computational results it also appears that contrary to the primal approach, the "'dual" approach is less influenced by scaling. Keywords: Fractional programming; Generalized fractional programming; Dinkelbach-type algorithms: Quasi- convexity; Karush-Kuhn-Tucker conditions: Duality 1. -
Glimpses Upon Quasiconvex Analysis Jean-Paul Penot
Glimpses upon quasiconvex analysis Jean-Paul Penot To cite this version: Jean-Paul Penot. Glimpses upon quasiconvex analysis. 2007. hal-00175200 HAL Id: hal-00175200 https://hal.archives-ouvertes.fr/hal-00175200 Preprint submitted on 27 Sep 2007 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. ESAIM: PROCEEDINGS, Vol. ?, 2007, 1-10 Editors: Will be set by the publisher DOI: (will be inserted later) GLIMPSES UPON QUASICONVEX ANALYSIS Jean-Paul Penot Abstract. We review various sorts of generalized convexity and we raise some questions about them. We stress the importance of some special subclasses of quasiconvex functions. Dedicated to Marc Att´eia R´esum´e. Nous passons en revue quelques notions de convexit´eg´en´eralis´ee.Nous tentons de les relier et nous soulevons quelques questions. Nous soulignons l’importance de quelques classes particuli`eres de fonctions quasiconvexes. 1. Introduction Empires usually are well structured entities, with unified, strong rules (for instance, the length of axles of carts in the Chinese Empire and the Roman Empire, a crucial rule when building a road network). On the contrary, associated kingdoms may have diverging rules and uses. -
Disciplined Quasiconvex Programming Arxiv:1905.00562V3
Disciplined Quasiconvex Programming Akshay Agrawal Stephen Boyd [email protected] [email protected] March 2, 2020 Abstract We present a composition rule involving quasiconvex functions that general- izes the classical composition rule for convex functions. This rule complements well-known rules for the curvature of quasiconvex functions under increasing functions and pointwise maximums. We refer to the class of optimization problems generated by these rules, along with a base set of quasiconvex and quasiconcave functions, as disciplined quasiconvex programs. Disciplined qua- siconvex programming generalizes disciplined convex programming, the class of optimization problems targeted by most modern domain-specific languages for convex optimization. We describe an implementation of disciplined quasi- convex programming that makes it possible to specify and solve quasiconvex programs in CVXPY 1.0. 1 Introduction A real-valued function f is quasiconvex if its domain C is convex, and for any α 2 R, its α-sublevel sets f x 2 C j f(x) ≤ α g are convex [BV04, x3.4]. A function f is quasiconcave if −f is quasiconvex, and it is quasilinear if it is both quasiconvex and quasiconcave. A quasiconvex program (QCP) is a mathematical optimization problem in which the objective is to minimize a quasiconvex function over a convex arXiv:1905.00562v3 [math.OC] 27 Feb 2020 set. Because every convex function is also quasiconvex, QCPs generalize convex programs. Though QCPs are in general nonconvex, many can nonetheless be solved efficiently by a bisection method that involves solving a sequence of convex programs [BV04, x4.2.5], or by subgradient methods [Kiw01; Kon03]. -
Pseudoinvex Functions on Riemannian Manifolds and Applications in Fractional Programming Problems
U.P.B. Sci. Bull., Series A, Vol. 80, Iss. 3, 2018 ISSN 1223-7027 PSEUDOINVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS AND APPLICATIONS IN FRACTIONAL PROGRAMMING PROBLEMS Avanish Shahi1, S.K. Mishra2 In this paper, we have obtained pseudoinvex functions from the ratio of invex and related functions to an affine and some generalized invex functions on Rie- mannian manifolds. Further, we establish sufficient optimality conditions and duality theorems for fractional nonlinear optimization problems under weaker assumptions on Riemannian manifolds. Keywords: Riemannian manifolds, fractional programming problems, pseudoinvex func- tions, optimality conditions, duality MSC2010: 58D 17; 90C 30; 90C 32; 90C 46 1. Introduction The ratio of a convex function to a positive affine function is a pseudoconvex function, see, Cambini and Martein [2]. The concept of invex and generalized invex functions was given by Hanson [8] and the same on Riemannian manifolds was given by Pini [14] and Barani and Pouryayevali [1]. Now, we extend the results of Cambini and Martein [2] to Riemannian manifolds. In our case the ratio of an invex function to a positive affine function is a pseudoinvex function and some other similar results also. A nonlinear fractional programming problem is an optimization problem. In the applications of fractional programming the quotient of two functions is to be maximized or minimized. If f is convex, g is concave and h is convex then the fractional programming is defined as convex-concave fractional programming problem. If all the functions f; g and h are invex then the programming is known as an invex problem. For invex fractional programming problems, we may cited the fundamental work of Craven [4] in which the invex function was advised first time and also the work of Reddy and Mukherjee [17], Singh and Hanson [18], Mishra and Giorgio [13], Craven [3] and Craven and Glover [5]. -
Disciplined Quasiconvex Programming
Optimization Letters (2020) 14:1643–1657 https://doi.org/10.1007/s11590-020-01561-8 ORIGINAL PAPER Disciplined quasiconvex programming Akshay Agrawal1 · Stephen Boyd1 Received: 25 September 2019 / Accepted: 22 February 2020 / Published online: 2 March 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020 Abstract We present a composition rule involving quasiconvex functions that generalizes the classical composition rule for convex functions. This rule complements well-known rules for the curvature of quasiconvex functions under increasing functions and point- wise maximums. We refer to the class of optimization problems generated by these rules, along with a base set of quasiconvex and quasiconcave functions, as disciplined quasiconvex programs. Disciplined quasiconvex programming generalizes disciplined convex programming, the class of optimization problems targeted by most modern domain-specific languages for convex optimization. We describe an implementation of disciplined quasiconvex programming that makes it possible to specify and solve quasiconvex programs in CVXPY 1.0. Keywords Quasiconvex programming · Convex optimization · Domain-specific languages 1 Introduction A real-valued function f is quasiconvex if its domain C is convex, and for any α ∈ R, its α-sublevel sets { x ∈ C | f (x) ≤ α } are convex [5, §3.4]. A function f is quasiconcave if − f is quasiconvex, and it is quasilinear if it is both quasiconvex and quasiconcave. A quasiconvex program (QCP) is a mathematical optimization problem in which the objective is to minimize a quasiconvex function over a convex set. Because every convex function is also quasiconvex, QCPs generalize convex programs. Though QCPs are in general nonconvex, many can nonetheless be solved efficiently by a bisection method that involves solving a sequence of convex programs [5, §4.2.5], or by subgradient methods [21,22].