Status determination by interior-point methods for convex optimization problems in domain-driven form

Abstract

We study the geometry of convex optimization problems given in a Domain-Driven form and categorize possible statuses of these problems using duality theory. Our duality theory for the Domain-Driven form, which accepts both conic and non-conic constraints, lets us determine and certify statuses of a problem as rigorously as the best approaches for conic formulations (which have been demonstrably very efficient in this context). We analyze the performance of a class of infeasible-start primal-dual algorithms for the Domain-Driven form in returning the certificates for the defined statuses. Our iteration complexity bounds for this more practical Domain-Driven form match the best ones available for conic formulations. At the end, we propose some stopping criteria for practical algorithms based on insights gained from our analyses.

A Algorithm for the Domain-Driven form

In this section, we describe a family of predictor-corrector algorithms for the general Domain-Driven form. The main part of the algorithm is a linear system we solve at every iteration. Let F be a matrix whose rows give a basis for the kernel of $A^\top$ and let $c_A$ be any vector such that $A^\top c_A=c$. Let us define:

$$\begin{aligned} U:=\left[ \begin{array}{ccc} A &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 \\ 0 &{} -c_A &{} -F^\top \\ c^\top &{} 0 &{} 0 \end{array} \right] , \ \ \ \ r^0:= \left[ \begin{array}{c} -A^\top y^0 - c \\ -y_{\tau ,0}+ \langle c_A, z^0\rangle \\ Fz^0 \end{array} \right] . \end{aligned}$$

(92)

At a current point $(x,\tau ,y)$, we construct the following linear system

$$\begin{aligned} U^\top \underbrace{ \left[ \begin{array}{cc} {\bar{H}}(x,\tau ) &{} 0 \\ 0 &{} \left[ {\hat{H}}(x,\tau ,y) \right] ^{-1} \end{array} \right] }_{{\mathcal {H}}({\bar{H}}, {\hat{H}})} U \left[ \begin{array}{c} {\bar{d}}_x \\ d_\tau \\ d_v \end{array}\right] =r_{RHS}, \nonumber \\ d_x:={\bar{d}}_x-d_\tau x, \ \ \ \ \ d_y:=-d_\tau c_A - F^\top d_v, \end{aligned}$$

(93)

where ${\bar{H}}(x,\tau )$ and ${\hat{H}}(x,\tau ,y)$ are positive definite matrices based on the Hessians of the s.c. functions. ${\bar{H}}(x,\tau )$ is obtained from $\frac{1}{\tau ^2} \varPhi ''(Ax+\frac{1}{\tau } z^0)$ by adding a row and column as given in [9]-(38). ${\hat{H}}(x,\tau ,y)$ can be chosen from a range of matrices close to $\mu ^2 {\bar{H}}(x,\tau )$ in the sense of [9]-(47).

For a point $(x,\tau ,y) \in Q_{DD}$, defined in (11), we define a proximity measure as

$$\begin{aligned} \varOmega _\mu (x,\tau ,y):= & {} \varPhi \left( Ax+\frac{1}{\tau } z^0 \right) + \varPhi _* \left( \frac{\tau y}{\mu } \right) - \frac{\tau }{\mu } \langle y , A x+ \frac{1}{\tau }z^0 \rangle , \\ \mu:= & {} \mu (x,\tau ,y), \ \ \text {as was given in}~{(12)}. \end{aligned}$$

For the statement of the algorithm, we also need the following vector for linearizing $\varOmega _\mu$.

$$\begin{aligned} \psi ^c:=\left[ \begin{array}{c}\frac{1}{\tau } \varPhi ' \\ -\frac{1}{\tau } \langle \varPhi ' ,Ax+\frac{1}{\tau }z^0 \rangle +\frac{1}{\mu } \langle y, \varPhi _*' \rangle + \frac{1}{\mu } (y_{\tau ,0}+\tau \langle c,x \rangle )\\ \frac{ \tau }{\mu } \varPhi _*' \\ \frac{\tau }{\mu } \end{array}\right] . \end{aligned}$$

It is proved in [9] that for suitable choices of $\xi$, $\delta _1$, and $\delta _2$, PCA$(\xi ,\delta _1,\delta _2)$ becomes a polynomial-time predictor-corrector algorithm (PtPCA):

Theorem 4

For the polynomial-time predictor-corrector algorithm (PtPCA), there exists a positive absolute constant $\gamma$ depending on $\xi$ such that after N iterations, the algorithm returns a point $(x,\tau ,y) \in Q_{DD}$ close to the central path that satisfies

$$\begin{aligned} \mu (x,\tau ,y) \ge \exp \left( \frac{\gamma }{\sqrt{\vartheta }} N \right) . \end{aligned}$$