Constrained array optimization by penalty function techniques

We investigate a stochastic penalty algorithm, which can be used to find a constrained optimum point for a concave or convex objective function subject to a nonlinear constraint which forms a connected region, even when we do not have the objective function available, but have only a noisy estimate of the objective function. When the constraint consists of one linear equation, we prove convergence to the constrained optimum value and bound the rate of convergence of the algorithm to the constrained optimum value. We then apply this algorithm to the nonlinear problem of automatically making an array of detectors form a beam in a desired direction in space when unknown interfering noise is present so as to maximize the signal-to-noise ratio subject to a constraint on the super-gain ratio.