Projections onto the canonical simplex with additional linear inequalities
For researchers in optimization and robust optimization, this provides efficient algorithms for a specific projection problem, but the contribution is incremental as it extends existing projection techniques.
The paper tackles distributionally robust optimization by reducing it to projections onto the canonical simplex with additional linear inequalities. It proposes optimization methods with guaranteed convergence and demonstrates (almost) linear observed complexity.
We consider the distributionally robust optimization and show that computing the distributional worst-case is equivalent to computing the projection onto the canonical simplex with additional linear inequality. We consider several distance functions to measure the distance of distributions. We write the projections as optimization problems and show that they are equivalent to finding a zero of real-valued functions. We prove that these functions possess nice properties such as monotonicity or convexity. We design optimization methods with guaranteed convergence and derive their theoretical complexity. We demonstrate that our methods have (almost) linear observed complexity.