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This paper considers the problem of minimizing an expectation function over a closed convex set, coupled with a {color{black} functional or expectation} constraint on either decision variables or problem parameters. We first present a new stochastic approximation (SA) type algorithm, namely the cooperative SA (CSA), to handle problems with the constraint on devision variables. We show that this algorithm exhibits the optimal ${cal O}(1/epsilon^2)$ rate of convergence, in terms of both optimality gap and constraint violation, when the objective and constraint functions are generally convex, where $epsilon$ denotes the optimality gap and infeasibility. Moreover, we show that this rate of convergence can be improved to ${cal O}(1/epsilon)$ if the objective and constraint functions are strongly convex. We then present a variant of CSA, namely the cooperative stochastic parameter approximation (CSPA) algorithm, to deal with the situation when the constraint is defined over problem parameters and show that it exhibits similar optimal rate of convergence to CSA. It is worth noting that CSA and CSPA are primal methods which do not require the iterations on the dual space and/or the estimation on the size of the dual variables. To the best of our knowledge, this is the first time that such optimal SA methods for solving functional or expectation constrained stochastic optimization are presented in the literature.
This paper considers the problem of minimizing a convex expectation function with a set of inequality convex expectation constraints. We present a computable stochastic approximation type algorithm, namely the stochastic linearized proximal method of
Stochastic convex optimization problems with expectation constraints (SOECs) are encountered in statistics and machine learning, business, and engineering. In data-rich environments, the SOEC objective and constraints contain expectations defined wit
In this paper, we focus on the problem of stochastic optimization where the objective function can be written as an expectation function over a closed convex set. We also consider multiple expectation constraints which restrict the domain of the prob
In this paper, we address a variant of the marketing mix optimization (MMO) problem which is commonly encountered in many industries, e.g., retail and consumer packaged goods (CPG) industries. This problem requires the spend for each marketing activi
In this paper, we consider a stochastic distributed nonconvex optimization problem with the cost function being distributed over $n$ agents having access only to zeroth-order (ZO) information of the cost. This problem has various machine learning app