Optimization of a convex program with a polynomial perturbation

Ravi Kannan, Luis Rademacher

Abstract: We consider the problem of minimizing a convex function plus a polynomial p<math><mi is="true">p</mi></math> over a convex body K<math><mi is="true">K</mi></math>. We give an algorithm that outputs a solution x<math><mi is="true">x</mi></math> whose value is within ϵrangeK(p)<math><mi is="true">ϵ</mi><msub is="true"><mrow is="true"><mo is="true">range</mo></mrow><mrow is="true"><mi is="true">K</mi></mrow></msub><mrow is="true"><mo is="true">(</mo><mi is="true">p</mi><mo is="true">)</mo></mrow></math> of the optimum value, where rangeK(p)=supx∈Kp(x)−infx∈Kp(x)<math><msub is="true"><mrow is="true"><mo is="true">range</mo></mrow><mrow is="true"><mi is="true">K</mi></mrow></msub><mrow is="true"><mo is="true">(</mo><mi is="true">p</mi><mo is="true">)</mo></mrow><mo is="true">=</mo><msub is="true"><mrow is="true"><mo is="true">sup</mo></mrow><mrow is="true"><mi is="true">x</mi><mo is="true">∈</mo><mi is="true">K</mi></mrow></msub><mi is="true">p</mi><mrow is="true"><mo is="true">(</mo><mi is="true">x</mi><mo is="true">)</mo></mrow><mo is="true">−</mo><msub is="true"><mrow is="true"><mo is="true">inf</mo></mrow><mrow is="true"><mi is="true">x</mi><mo is="true">∈</mo><mi is="true">K</mi></mrow></msub><mi is="true">p</mi><mrow is="true"><mo is="true">(</mo><mi is="true">x</mi><mo is="true">)</mo></mrow></math>. When p<math><mi is="true">p</mi></math> depends only on a constant number of variables, the algorithm runs in time polynomial in 1/ϵ<math><mn is="true">1</mn><mo is="true">/</mo><mi is="true">ϵ</mi></math>, the degree of p<math><mi is="true">p</mi></math>, the time to round K<math><mi is="true">K</mi></math> and the time to solve the convex program that results by setting p=0<math><mi is="true">p</mi><mo is="true">=</mo><mn is="true">0</mn></math>.