An interior-point method for fractional programs with convex constraints
We present an interior point method for a class of fractional programming problems with convex constraints. The problems under consideration can be reduced to the following standard form: minimize {l_brace}{sub {beta}}{sup {alpha}}{vert_bar} f{sub i} (X, {alpha}, {beta}) {<=} 0 for 1 {<=} i {<=} m{r_brace}, where x {element_of} R{sup n}, {alpha}, {beta} {element_of} R, {beta} {<=} 0 and f{sub i} are certain (convex) functions of {alpha}, {beta} and x. The rate of convergence for our algorithm is polynomial, just as in the case of interior-point methods for solving convex programs, even though the objective function {open_quotes}{sub {beta}}{sup {alpha}}{close_quotes} for this problem is not convex. In our approach, the rate of convergence is measured in terms of the volume of suitably constructed two-dimensional convex sets R{sub k} containing the optimal point, and the area of R{sub k} is reduced by a factor of O(1-1/{radical}{var_theta}) at each iteration. Here, {var_theta} is a self-concordance parameter of the logarithmic barrier function -{Sigma}{sub i=1}{sup m} log (-f{sub i}({alpha}, {beta}, x)) and is typically proportional to the number m of convex constraints. An application of this algorithm is the minimization of the condition number of certain matrices A(x) that depend affinely on some control vector x.
- OSTI ID:
- 36163
- Report Number(s):
- CONF-9408161--
- Country of Publication:
- United States
- Language:
- English
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