Nonsmooth projection-free optimization with functional constraints
提出一种基于次梯度的算法，用于处理带一般凸函数不等式约束的非光滑凸优化问题，无需向可行集投影，每次迭代只需一次线性最小化调用和一次次梯度计算，达到epsilon-次优解需O(epsilon^{-2})次迭代。
Abstract This paper presents a subgradient-based algorithm for constrained nonsmooth convex optimization that does not require projections onto the feasible set. While the well-established Frank–Wolfe algorithm and its variants already avoid projections, they are primarily designed for smooth objective functions. In contrast, our proposed algorithm can handle nonsmooth problems with general convex functional inequality constraints. It achieves an $$\epsilon $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ϵ</mml:mi> </mml:math> -suboptimal solution in $$\mathcal {O}(\epsilon ^{-2})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>ϵ</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> iterations, with each iteration requiring only a single (potentially inexact) Linear Minimization Oracle call and a (possibly inexact) subgradient computation. This performance is consistent with existing lower bounds. Similar performance is observed when deterministic subgradients are replaced with stochastic subgradients. In the special case where there are no functional inequality constraints, our algorithm competes favorably with a recent nonsmooth projection-free method designed for constraint-free problems. Our approach utilizes a simple separation scheme in conjunction with a new Lagrange multiplier update rule.