A stochastic primal-dual method for a class of nonconvex constrained optimization

Abstract

In this paper we study a class of nonconvex optimization which involves uncertainty in the objective and a large number of nonconvex functional constraints. Challenges often arise when solving this type of problems due to the nonconvexity of the feasible set and the high cost of calculating function value and gradient of all constraints simultaneously. To handle these issues, we propose a stochastic primal-dual method in this paper. At each iteration, a proximal subproblem based on a stochastic approximation to an augmented Lagrangian function is solved to update the primal variable, which is then used to update dual variables. We explore theoretical properties of the proposed algorithm and establish its iteration and sample complexities to find an $\epsilon$-stationary point of the original problem. Numerical tests on a weighted maximin dispersion problem and a nonconvex quadratically constrained optimization problem demonstrate the promising performance of the proposed algorithm.

Appendix

1.1 A.1: Lipschitz-continuity of $\psi _{\beta }\left( u,v\right)$ in $v \in \mathbb {R}_{+}$

For any $u \in \mathbb {R}$,

$$\begin{aligned} \left| \psi _{\beta }\left( u,v_{1}\right) -\psi _{\beta }\left( u,v_{2}\right) \right| \le \left| u \right| \left| v_{1}-v_{2} \right| ,\quad \forall v_{1},v_{2}\ge 0 . \end{aligned}$$

Proof

For $\psi _{\beta }\left( u, v\right) =\left\{ \begin{array}{ll} u v+\frac{\beta }{2} u^{2}, &{} \text{ if } \beta u+v \ge 0 \\ -\frac{v^{2}}{2 \beta }, &{} \text{ if } \beta u+v<0\end{array}\right. ,\beta >0$, the result obviously holds for any $u \ge 0$. So we will just talk about case when $u < 0$.

If $\beta u+v_{1} \ge 0,\beta u+v_{2} \ge 0$, the result holds obviously. If $\beta u+v_{1} \ge 0,\beta u+v_{2} < 0$, since $\psi _{\beta }\left( u, v\right)$ is monotonically decreasing in $v\ge 0$ when $u < 0$, we have

$$\begin{aligned} \left| \psi _{\beta }\left( u,v_{1}\right) -\psi _{\beta }\left( u,v_{2}\right) \right|&=\psi _{\beta }\left( u,v_{2}\right) -\psi _{\beta }\left( u,v_{1}\right) = -\frac{v_{2}^{2}}{2 \beta }-u v_{1}-\frac{\beta }{2} u^{2} \\&\le u v_{2}+\frac{\beta }{2} u^{2} -u v_{1}-\frac{\beta }{2} u^{2} =\left| u \right| \left| v_{1}-v_{2} \right| . \end{aligned}$$

If $\beta u+v_{1}< 0,\beta u+v_{2} < 0$, then we have

$$\begin{aligned} \left| \psi _{\beta }\left( u,v_{1}\right) -\psi _{\beta }\left( u,v_{2}\right) \right| =\frac{ \left| v_{1}^{2}-v_{2}^{2} \right| }{2\beta }= \frac{ v_{1}+v_{2} }{2\beta } \left| v_{1}-v_{2} \right| \le -u \left| v_{1}-v_{2} \right| =\left| u \right| \left| v_{1}-v_{2} \right| . \end{aligned}$$

$\square$

1.2 A.2: Nonexpansiveness of prox operator

For $x^{k+1}$ defined in (15) and ${\hat{x}}^{k+1}$ defined in (30), it holds that

$$\begin{aligned} \left\| x^{k+1}-{\hat{x}}^{k+1} \right\| \le \alpha _{k} \left\| \nabla f_{0}\left( x^{k}\right) +\nabla _{x}\Psi _{\beta }\left( x^{k},z^{k}\right) -g_{0}^{k}-h^{k} \right\| . \end{aligned}$$

Proof

From the optimality condition for (15), there exists a subgradient $u \in \partial \chi _{0}\left( x^{k+1}\right)$ such that

$$\begin{aligned} \left\langle u+g_{0}^{k}+h^{k}+\frac{1}{\alpha _{k}}\nabla _{y} V\left( x^{k}, y\right) {|}_{y=x^{k+1}} ,x-x^{k+1} \right\rangle \ge 0,\quad \forall x\in X. \end{aligned}$$

Applying the convexity of $\chi _{0}$, we obtain that

$$\begin{aligned} \chi _{0}\left( x \right) -\chi _{0}\left( x^{k+1} \right) + \left\langle g_{0}^{k}+h^{k}+\frac{1}{\alpha _{k}}\nabla _{y} V\left( x^{k}, y\right) {|}_{y=x^{k+1}} ,x-x^{k+1}\right\rangle \ge 0,\quad \forall x\in X. \end{aligned}$$

(54)

Similarly, it yields from the optimality condition for ${\hat{x}}^{k+1}$ that for any $x\in X$,

$$\begin{aligned}&\chi _{0}\left( x \right) -\chi _{0}\left( {\hat{x}}^{k+1} \right) + \left\langle \nabla f_{0}\left( x^{k} \right) +\nabla _{x} \Psi _{\beta }\left( x^{k},z^{k}\right) \right. \nonumber \\&\quad \left. +\frac{1}{\alpha _{k}}\nabla _{y} V\left( x^{k}, y\right) {|}_{y={\hat{x}}^{k+1}} ,x-{\hat{x}}^{k+1} \right\rangle \ge 0. \end{aligned}$$

(55)

Letting $x={\hat{x}}^{k+1}$ in (54) and $x=x^{k+1}$ in (55) and adding them up, we obtain

$$\begin{aligned}&\, \left\langle \nabla f_{0}\left( x^{k}\right) +\nabla _{x}\Psi _{\beta }\left( x^{k},z^{k}\right) -g_{0}^{k}-h^{k}, x^{k+1}-{\hat{x}}^{k+1} \right\rangle \\ \ge&\, \frac{1}{\alpha _{k}} \left\langle \nabla _{y} V\left( x^{k},{y} \right) {|}_{y=x^{k+1}} -\nabla _{y} V\left( x^{k}, y\right) {|}_{y={\hat{x}}^{k+1}} ,x^{k+1}-{\hat{x}}^{k+1} \right\rangle \\ =&\, \frac{1}{\alpha _{k}} \left\langle \nabla v\left( x^{k+1}\right) -\nabla v\left( {\hat{x}}^{k+1}\right) ,x^{k+1}-{\hat{x}}^{k+1} \right\rangle \\ \ge&\frac{1}{\alpha _{k}} \left\| x^{k+1}-{\hat{x}}^{k+1} \right\| ^{2} , \end{aligned}$$

where the last inequality comes from the 1-strong convexity of $v\left( x\right)$. Then applying Cauchy-Schwarz inequality indicates the result. $\square$