Abstract
This paper presents a canonical dual approach to minimizing the sum of a quadratic function and the ratio of two quadratic functions, which is a type of non-convex optimization problem subject to an elliptic constraint. We first relax the fractional structure by introducing a family of parametric subproblems. Under proper conditions on the "problem-defining" matrices associated with the three quadratic functions, we show that the canonical dual of each subproblem becomes a one-dimensional concave maximization problem that exhibits no duality gap. Since the infimum of the optima of the parameterized subproblems leads to a solution to the original problem, we then derive some optimality conditions and existence conditions for finding a global minimizer of the original problem. Some numerical results using the quasi-Newton and line search methods are presented to illustrate our approach.
| Original language | English |
|---|---|
| Pages (from-to) | 337-353 |
| Number of pages | 17 |
| Journal | Journal of Global Optimization |
| Volume | 45 |
| Issue number | 3 |
| DOIs | |
| Publication status | Published - 2009 Nov 1 |
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All Science Journal Classification (ASJC) codes
- Computer Science Applications
- Control and Optimization
- Management Science and Operations Research
- Applied Mathematics
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Global optimization for a class of fractional programming problems. / Fang, Shu Cherng; Gao, David Y.; Sheu, Ruey-Lin; Xing, Wenxun.
In: Journal of Global Optimization, Vol. 45, No. 3, 01.11.2009, p. 337-353.Research output: Contribution to journal › Article
TY - JOUR
T1 - Global optimization for a class of fractional programming problems
AU - Fang, Shu Cherng
AU - Gao, David Y.
AU - Sheu, Ruey-Lin
AU - Xing, Wenxun
PY - 2009/11/1
Y1 - 2009/11/1
N2 - This paper presents a canonical dual approach to minimizing the sum of a quadratic function and the ratio of two quadratic functions, which is a type of non-convex optimization problem subject to an elliptic constraint. We first relax the fractional structure by introducing a family of parametric subproblems. Under proper conditions on the "problem-defining" matrices associated with the three quadratic functions, we show that the canonical dual of each subproblem becomes a one-dimensional concave maximization problem that exhibits no duality gap. Since the infimum of the optima of the parameterized subproblems leads to a solution to the original problem, we then derive some optimality conditions and existence conditions for finding a global minimizer of the original problem. Some numerical results using the quasi-Newton and line search methods are presented to illustrate our approach.
AB - This paper presents a canonical dual approach to minimizing the sum of a quadratic function and the ratio of two quadratic functions, which is a type of non-convex optimization problem subject to an elliptic constraint. We first relax the fractional structure by introducing a family of parametric subproblems. Under proper conditions on the "problem-defining" matrices associated with the three quadratic functions, we show that the canonical dual of each subproblem becomes a one-dimensional concave maximization problem that exhibits no duality gap. Since the infimum of the optima of the parameterized subproblems leads to a solution to the original problem, we then derive some optimality conditions and existence conditions for finding a global minimizer of the original problem. Some numerical results using the quasi-Newton and line search methods are presented to illustrate our approach.
UR - http://www.scopus.com/inward/record.url?scp=70349694498&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=70349694498&partnerID=8YFLogxK
U2 - 10.1007/s10898-008-9378-7
DO - 10.1007/s10898-008-9378-7
M3 - Article
VL - 45
SP - 337
EP - 353
JO - Journal of Global Optimization
JF - Journal of Global Optimization
SN - 0925-5001
IS - 3
ER -