Curvature integrals and iteration complexities in SDP and symmetric cone programs

Abstract

In this paper, we study iteration complexities of Mizuno-Todd-Ye predictor-corrector (MTY-PC) algorithms in SDP and symmetric cone programs by way of curvature integrals. The curvature integral is defined along the central path, reflecting the geometric structure of the central path. Integrating curvature along the central path, we obtain a precise estimate of the number of iterations to solve the problem. It has been shown for LP that the number of iterations is asymptotically precisely estimated with the integral divided by $\sqrt{\beta}$, where β is the opening parameter of the neighborhood of the central path in MTY-PC algorithms. Furthermore, this estimate agrees quite well with the observed number of iterations of the algorithm even when β is close to one and when applied to solve large LP instances from NETLIB. The purpose of this paper is to develop direct extensions of these two results to SDP and symmetric cone programs. More specifically, we give concrete formulas for curvature integrals in SDP and symmetric cone programs and give asymptotic estimates for iteration complexities. Through numerical experiments with large SDP instances from SDPLIB, we demonstrate that the number of iterations is explained quite well with the integral even for a large step size which is enough to solve practical large problems.

Appendix

In appendix, we lay out proofs of Lemma 3.5 and Lemma 3.7 used in the proof of Theorem 3.8.

Proof of Lemma 3.5

Let

$$ F_{w}\bigl(w'\bigr) = \begin{bmatrix} W_{P}(w') - W_{P}(w)\\ \mathcal {A}X' - b\\ \mathcal {A}^* y' + S' - C \end{bmatrix}. $$

Then, letting R=W _P (w)/ν, we have

$$ F_{w}(w) = 0\quad \text{and}\quad F_{w}(w_\nu) = \begin{bmatrix} \nu (I - R)\\ 0\\ 0 \end{bmatrix}. $$

Hence, from the continuity of F _w , it holds on the neighborhood of the central path that with constants C ₁,C ₂,

$$\begin{aligned} \lVert w - w_\nu\rVert =& C_1 \big\Vert F_{w}^{-1}(0) - F_{w}^{-1}\bigl(\bigl[\nu (I - R),0,0\bigr]^{T} \bigr)\big\Vert = C_2 \big\Vert \nu (I - R)\big\Vert _{F}\\ =& \mathcal {O}\bigl(\beta^2\bigr). \end{aligned}$$

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Proof of Lemma 3.7

Let

$$ \widetilde{\mathcal{A}} := \left ( \begin{array}{c@{\quad}c@{\quad}c} \mathcal{A} & 0 & 0\\ 0 & I & \mathcal{A}^*\\ E_{\nu} & F_{\nu} & 0 \end{array} \right ),\quad \widetilde{x} := \left [ \begin{array}{c} \operatorname {vec}(\Delta X_{\nu})\\ \operatorname {vec}(\Delta S_{\nu})\\ \Delta y_{\nu} \end{array} \right ],\quad \widetilde{b} := \left [ \begin{array}{c} 0\\ 0\\ \operatorname {vec}(\nu I) \end{array} \right ], $$

$$\begin{aligned} &\widetilde{\Delta \mathcal{A}} := \left ( \begin{array}{c@{\quad}c@{\quad}c} 0 & 0 & 0\\ 0 & 0 & 0\\ E - E_{\nu} & F - F_{\nu} & 0 \end{array} \right ),\quad \widetilde{\Delta x} := \left [ \begin{array}{c} \operatorname {vec}(\Delta X - \Delta X_{\nu})\\ \operatorname {vec}(\Delta S - \Delta S_{\nu})\\ \Delta y - \Delta y_{\nu} \end{array} \right ],\\ & \widetilde{\Delta b} := \left [ \begin{array}{c} 0\\ 0\\ \operatorname {vec}(\mathcal {H}_{p}(XS) - \nu I) \end{array} \right ], \end{aligned}$$

where

$$\begin{aligned} E_{\nu} :=& \frac{1}{2}\bigl(P_{\nu}^{-T}S_{\nu} \otimes P_{\nu} + P_{\nu}\otimes P_{\nu}^{-T}S_{\nu}\bigr), \quad F_{\nu} := \frac{1}{2}\bigl(P_{\nu}X_{\nu} \otimes P_{\nu}^{-T} + P_{\nu}^{-T}\otimes P_{\nu}X_{\nu} \bigr), \\ E :=& \frac{1}{2}\bigl(P^{-T}S\otimes P + P\otimes P^{-T}S\bigr),\quad F := \frac{1}{2}\bigl(PX\otimes P^{-T} + P^{-T}\otimes PX \bigr). \end{aligned}$$

Note that by conventions, we allow above formulations of operators and matrices. Then, by Lemma 3.6, we have

$$\begin{aligned} \lVert \widetilde{\Delta x}\rVert \leq \frac{\lVert \widetilde{\mathcal{A}}^{-1}\rVert }{1 - \lVert \widetilde{\mathcal{A}}^{-1}\rVert \lVert \widetilde{\Delta \mathcal{A}}\rVert } \bigl(\lVert \widetilde{\Delta b}\rVert + \lVert \widetilde{\Delta \mathcal{A}}\rVert \lVert \widetilde{x}\rVert \bigr). \end{aligned}$$

(71)

We first show that ${\lVert \widetilde{\Delta \mathcal{A}}\rVert }_{F} = \mathcal {O}(\beta^{2})$. We have

$$\begin{aligned} E - E_{\nu} =& \frac{1}{2} \bigl[\bigl(P^{-T}S\otimes P + P\otimes P^{-T}S\bigr) - \bigl(P_{\nu}^{-T}S_{\nu} \otimes P_{\nu} + P_{\nu}\otimes P_{\nu}^{-T}S_{\nu}\bigr)\bigr] \\ =& \frac{1}{2} \bigl[\bigl(P^{-T}S\otimes P - P_{\nu}^{-T}S_{\nu} \otimes P_{\nu}\bigr) + \bigl( P\otimes P^{-T}S - P_{\nu} \otimes P_{\nu}^{-T}S_{\nu}\bigr)\bigr] \\ =& \frac{1}{2} \bigl[\bigl(\bigl(P^{-T}-P_{\nu}^{-T} \bigr)S \otimes P + P_{\nu}^{-T} (S - S_{\nu}) \otimes P + \bigl(P_{\nu}^{-T} S_{\nu} \otimes (P - P_{\nu})\bigr) \bigr)\\ &{} + \bigl((P - P_{\nu})\otimes P^{-T}S + P_{\nu} \otimes \bigl(P^{-T}-P_{\nu}^{-T}\bigr)S + P_{\nu} \otimes P_{\nu}^{-T} (S - S_{\nu})\bigr) \bigr]. \end{aligned}$$

Since ∥S∥ _F , ∥S _ν ∥ _F , ∥P∥ _F , ∥P _ν ∥ _F are bounded on $\mathcal {F}$, and there exists a constant C such that ∥P−P _ν ∥ _F =C∥w−w _c ∥ _F and ${\lVert w - w_{c}\rVert }_{F} = \mathcal {O}(\beta^{2})$ from Lemma 3.5, it follows that

$$ {\lVert E - E_{\nu}\rVert }_{F} = \mathcal {O}\bigl(\beta^2\bigr). $$

Similarly, we have ${\lVert F - F_{\nu}\rVert }_{F} = \mathcal {O}(\beta^{2})$. Thus we obtain

$$ {\lVert \widetilde{\Delta \mathcal{A}}\rVert }_{F} = \mathcal {O}\bigl(\beta^2\bigr). $$

Next, we show $\lVert \widetilde{\Delta b}\rVert = \mathcal {O}(\beta^{2})$. Similarly, it is easy to see that

$$\begin{aligned} \big\Vert \mathcal {H}_p(XS) - \nu I\big\Vert _{F} =& \frac{1}{2}\big\Vert \bigl(PXSP^{-1} - \nu I\bigr) + \bigl(P^{-T}SXP^{T} - \nu I\bigr)\big\Vert _{F} \\ \leq& \frac{1}{2} \bigl[ \big\Vert PX^{1/2}\bigl(X^{1/2}SX^{1/2} - \nu I\bigr)X^{-1/2}P^{-1}\big\Vert _{F} \\ &{}+ \big\Vert P^{-T}X^{-1/2}\bigl(X^{1/2}SX^{1/2} - \nu I\bigr)X^{1/2}P^{T}\big\Vert _{F} \bigr] \\ \leq& \frac{1}{2} \bigl[ {\lVert P\rVert }_{F} \big\Vert X^{1/2}\big\Vert _{F} \big\Vert X^{1/2}SX^{1/2} - \nu I\big\Vert _{F} \big\Vert X^{-1/2}\big\Vert _{F} \big\Vert P^{-1}\big\Vert _{F} \\ &{}+ \big\Vert P^{-T}\big\Vert _{F} \big\Vert X^{-1/2}\big\Vert _{F} \big\Vert X^{1/2}SX^{1/2} - \nu I\big\Vert _{F} \big\Vert X^{1/2}\big\Vert _{F} \big\Vert P^{T}\big\Vert _{F} \bigr]\\ =& \mathcal{O} \bigl(\beta^2\bigr). \end{aligned}$$

The last equality comes from the fact that ${\lVert X^{1/2}SX^{1/2} - \nu I\rVert }_{F} = \mathcal {O}(\beta^{2})$ and ∥P∥ _F and ∥X ^−1/2∥ _F are bounded on $\mathcal {F}$.

Since $\lVert \widetilde{\mathcal{A}}^{-1}\rVert$, $\lVert \widetilde{x}\rVert$ are bounded, it follows from above discussion that $\lVert \widetilde{\Delta x}\rVert = \mathcal {O}(\beta^{2})$, which implies that

$$ \big\Vert (\Delta X,\Delta y,\Delta S) - (\Delta X_{\nu}, \Delta y_{\nu},\Delta S_{\nu})\big\Vert _{F} = \mathcal{O}\bigl(\beta^2 \bigr). $$

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