Fast bundle-level methods for unconstrained and ball-constrained convex optimization

Abstract

In this paper, we study a special class of first-order methods, namely bundle-level (BL) type methods, which can utilize historical first-order information through cutting plane models to accelerate the solutions in practice. Recently, it has been shown in Lan (149(1–2):1–45, 2015) that an accelerated prox-level (APL) method and its variant, the uniform smoothing level (USL) method, have optimal iteration complexity for solving black-box and structured convex programming (CP) problems without requiring input of any smoothness information. However, these algorithms require the assumption on the boundedness of the feasible set and their efficiency relies on the solutions of two involved subproblems. Some other variants of BL methods which could handle unbounded feasible set have no iteration complexity provided. In this work we develop the fast APL (FAPL) method and fast USL (FUSL) method that can significantly improve the practical performance of the APL and USL methods in terms of both computational time and solution quality. Both FAPL and FUSL enjoy the same optimal iteration complexity as APL and USL, while the number of subproblems in each iteration is reduced from two to one, and an exact method is presented to solve the only subproblem in these algorithms. Furthermore, we introduce a generic algorithmic framework to solve unconstrained CP problems through solutions to a series of ball-constrained CP problems that also exhibits optimal iteration complexity. Our numerical results on solving some large-scale least squares problems and total variation based image reconstructions have shown advantages of these new BL type methods over APL, USL, and some other first-order methods.

Appendix A. Solving the subproblems of FAPL and FUSL

In this section, we introduce an efficient method to solve the subproblems (2.6) in the FAPL and FUSL methods, which are given in the form of

$$\begin{aligned} x_c^{*}:=\mathrm{argmin}_{x\in Q}\frac{1}{2}\Vert x-p\Vert ^2. \end{aligned}$$

(A.1)

Here, Q is a closed polyhedral set described by m linear inequalities, i.e.,

$$\begin{aligned} Q:=\{x\in \mathbb {R}^n: \left\langle A_i,x\right\rangle \le b_i, \ i=1,2,\ldots ,m\}, \end{aligned}$$

where $A_i\in \mathbb {R}^n$ and $b_i\in \mathbb {R}$ for $1\le i\le m$.

Now let us examine the Lagrange dual of (A.1) given by

$$\begin{aligned} \max _{\lambda \ge 0} \min _{x \in \mathbb {R}^n} \frac{1}{2}\Vert x-p\Vert ^2+\sum _{i=1}^{m}\lambda _i[\left\langle A_{i},x\right\rangle -b_i]. \end{aligned}$$

(A.2)

It can be checked from the theorem of alternatives that problem (A.2) is solvable if and only if $Q \ne \emptyset$. Indeed, if $Q \ne \emptyset$, it is obvious that the optimal value of (A.2) is finite. On the other hand, if $Q = \emptyset$, then there exists $\bar{\lambda } \ge 0$ such that $\bar{\lambda }^T A = 0$ and $\bar{\lambda }^T b < 0$, which implies that the optimal value of (A.2) goes to infinity. Moreover, if (A.2) is solvable and $\lambda ^*$ is one of its optimal dual solutions, then

$$\begin{aligned} x_c^{*}=p-\sum \limits _{i=1}^{m}\lambda _i^{*}A_i. \end{aligned}$$

(A.3)

It can also be easily seen that (A.2) is equivalent to

$$\begin{aligned} \max _{\lambda \ge 0} - \frac{1}{2}\lambda ^T M \lambda + C^T\lambda , \end{aligned}$$

(A.4)

where $M_{ij}:=\left\langle A_i,A_j\right\rangle ,\ C_i:=\left\langle A_i,p\right\rangle -b_i, \ \forall i,j=1,2,\ldots ,m.$ Hence, we can determine the feasibility of (A.1) or compute its optimal solution by solving the relatively simple problem (A.4).

Many algorithms are capable of solving the above nonnegative quadratic programming in (A.4) efficiently. Due to its low dimension (usually less than 10 in our practice), we propose a brute-force method to compute the exact solution of this problem. Consider the Lagrange dual associated with (A.4):

$$\begin{aligned} \min _{\lambda \ge 0}\max _{\mu \ge 0 }\mathcal {L}(\lambda ,\mu ):= \frac{1}{2}\lambda ^T M \lambda - (C^T+\mu )\lambda , \end{aligned}$$

where the dual variable is $\mu :=(\mu _1,\mu _2,\ldots ,\mu _m)$. Applying the KKT condition, we can see that $\lambda ^{*}\ge 0$ is a solution to problem (A.4) if and only if there exists $\mu ^*\ge 0$ such that

$$\begin{aligned} \nabla _{\lambda } \mathcal {L}(\lambda ^{*},\mu ^{*})=0 \ \ \ \text{ and } \ \ \ \langle \lambda ^*, \mu ^*\rangle =0. \end{aligned}$$

(A.5)

Note that the first identity in (A.5) is equivalent to a linear system:

$$\begin{aligned} \begin{pmatrix} M&-I\end{pmatrix}\begin{pmatrix}\lambda _1\\ \vdots \\ \lambda _m\\ \mu _1\\ \vdots \\ \mu _m\end{pmatrix}=\begin{pmatrix}C_1\\ C_2\\ \vdots \\ C_m \end{pmatrix}, \end{aligned}$$

(A.6)

where I is the $m\times m$ identity matrix. The above linear system has 2m variables and m equations. But for any $i=1,\ldots ,m$, we have either $\lambda _i=0$ or $\mu _i=0$, and hence we only need to consider $2^m$ possible cases on the non-negativity of these variables. Since m is rather small in practice, it is possible to exhaust all these $2^m$ cases to find the exact solution to (A.5). For each case, we first remove the m columns in the matrix $(M \ -I)$ which correspond to the m variables assumed to be 0, and then solve the remaining determined linear system. If all variables of the computed solution are non-negative, then solution $(\lambda ^{*},\mu ^{*})$ to (A.5) is found, and the exact solution $x_c^*$ to (A.1) is computed by (A.3), otherwise, we continue to examine the next case. It is interesting to observe that these different cases can also be considered in parallel to take the advantages of high performance computing techniques.