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We propose randomized subspace gradient methods for high-dimensionalconstrained optimization. While there have been similarly purposed studies onunconstrained optimization problems, there have been few on constrainedoptimization problems due to the difficulty of handling constraints. Ouralgorithms project gradient vectors onto a subspace that is a random projectionof the subspace spanned by the gradients of active constraints. We determinethe worst-case iteration complexity under linear and nonlinear settings andtheoretically confirm that our algorithms can take a larger step size thantheir deterministic version. From the advantages of taking longer step andrandomized subspace gradients, we show that our algorithms are especiallyefficient in view of time complexity when gradients cannot be obtained easily.Numerical experiments show that they tend to find better solutions because ofthe randomness of their subspace selection. Furthermore, they performs well incases where gradients could not be obtained directly, and instead, gradientsare obtained using directional derivatives.

Randomized subspace gradient method for constrained optimization