Multiparameter long-step interior methods for convex problems
Let G {contained_in} R{sup n} be a closed convex domain with known interior point {bar x}. Consider the problem P : minimize c{sup T} xs.t. x {element_of} G{sub f}, where G{sub f} = {r_brace}x {element_of} G {vert_bar} f{sup T} x {<=} 0{r_brace}; the setting covers, in particular, the case of infeasible start {bar x} int G{sub f}. We demonstrate that numerous developed so far path-following strategies for solving P (the standard path-following scheme with feasible start, the {open_quotes}big M{close_quotes} scheme with unfeasible start, etc.) can be treated as particular policies of tracing the surface of analytic centers x(t) = argmin[-ln(t{sub c}-c{sup T} x)-ln(t{sub f} - f{sup T} x)-ln(t{sub d}-d{sup T}x) + {var_theta}{sup -1}F(x)], where t = (t{sub c}, t{sub f}, t{sub d}) is the vector of parameters. F is a {open_quotes}good{close_quotes} ({var_theta}-self-concordant) barrier for G and d is defined by the requirement that the surface passes through {bar x}. We describe general predictor-corrector scheme for tracing surfaces of the above type, with emphasis on implementing {open_quotes}long steps{close_quotes} along the surface, indicate the relevant complexity results and present polynomial time applications of the scheme to a number of classes of convex problems (LP, QP, Geometrical and Semidefinite Programming, etc.). We present also polynomial time extensions for certain quasiconvex problems, e.g., the Generalized Eigenvalue one.
- OSTI ID:
- 36338
- Report Number(s):
- CONF-9408161--
- Country of Publication:
- United States
- Language:
- English
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