Condition-Based Complexity of Convex Optimization in Conic Linear Form via the Ellipsoid Algorithm

• Authors: Robert M. Freund;Jorge R. Vera
• Venue: SIAM Journal on Optimization
• Year: 1999

A convex optimization problem in conic linear form is an optimization problem of the form $$\begin{array}{lclr} CP(d): & {\rm maximize} & c^{T}x \\ & \mbox{ s. t. } & b-Ax \in C_{Y},\\ & & x \in C_{X} , \\ \end{array}$$ where $C_{X}$ and $C_{Y}$ are closed convex cones in $n$- and $m$-dimensional spaces $X$ and $Y,$ respectively, and the data for the system is $d=(A,b,c)$. We show that there is a version of the ellipsoid algorithm that can be applied to find an $\epsilon$-optimal solution of $CP(d)$ in at most $O(n^2\ln (\frac{{\cal C}(d)\|c\|_*}{c_1\epsilon}))$ iterations of the ellipsoid algorithm, where each iteration must either perform a separation cut on one of the cones $C_X$ or $C_Y$ or perform a related optimality cut. The quantity ${\cal C}(d)$ is the ``condition number" of the program $CP(d)$ originally developed by Renegar and is essentially a scale-invariant reciprocal of the smallest data perturbation $\Delta d=(\Delta A,\Delta b, \Delta c)$ for which the system $CP(d+\Delta d)$ becomes either infeasible or unbounded. The scalar quantity $c_1$ is a constant that depends only on the simple notion of the ``width" of the cones and is independent of the problem data $d=(A,b,c)$ but may depend on the dimensions $m$ and/or $n$.