Sequential Quadratic Optimization for Nonlinear Equality Constrained Stochastic Optimization
This work addresses optimization problems with stochastic objectives for researchers and practitioners in fields like machine learning and operations research, but it is incremental as it builds on existing SQP methods.
The authors tackled the problem of solving smooth nonlinear optimization with equality constraints where the objective function is stochastic and intractable to compute directly, proposing sequential quadratic optimization algorithms. They proved convergence from remote starting points for both deterministic and stochastic versions, with numerical experiments demonstrating practical performance.
Sequential quadratic optimization algorithms are proposed for solving smooth nonlinear optimization problems with equality constraints. The main focus is an algorithm proposed for the case when the constraint functions are deterministic, and constraint function and derivative values can be computed explicitly, but the objective function is stochastic. It is assumed in this setting that it is intractable to compute objective function and derivative values explicitly, although one can compute stochastic function and gradient estimates. As a starting point for this stochastic setting, an algorithm is proposed for the deterministic setting that is modeled after a state-of-the-art line-search SQP algorithm, but uses a stepsize selection scheme based on Lipschitz constants (or adaptively estimated Lipschitz constants) in place of the line search. This sets the stage for the proposed algorithm for the stochastic setting, for which it is assumed that line searches would be intractable. Under reasonable assumptions, convergence (resp.,~convergence in expectation) from remote starting points is proved for the proposed deterministic (resp.,~stochastic) algorithm. The results of numerical experiments demonstrate the practical performance of our proposed techniques.