Fast projection onto the intersection of simplex and singly linear constraint and its generalized Jacobian

Solving the distributional worst-case in the distributionally robust optimization problem is equivalent to finding the projection onto the intersection of simplex and singly linear inequality constraint. This projection is a key component in the design of efficient first-order algorithms. This paper focuses on developing efficient algorithms for computing the projection onto the intersection of simplex and singly linear inequality constraint. Based on the Lagrangian duality theory, the studied projection can be obtained by solving a univariate nonsmooth equation. We employ an algorithm called LRSA, which leverages the Lagrangian duality approach and the secant method to compute this projection. In this algorithm, a modified secant method is specifically designed to solve the piecewise linear equation. Additionally, due to semismoothness of the resulting equation, the semismooth Newton (SSN) method is a natural choice for solving it. Numerical experiments demonstrate that LRSA outperforms SSN algorithm and the state-of-the-art optimization solver called Gurobi. Moreover, we derive explicit formulas for the generalized HS-Jacobian of the projection, which are essential for designing second-order nonsmooth Newton algorithms.