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This paper considers the problem of minimizing a convex expectation function with a set of inequality convex expectation constraints. We propose a stochastic augmented Lagrangian-type algorithm—namely, the stochastic linearized proximal method of multipliers—to solve this convex stochastic optimization problem. This algorithm can be roughly viewed as a hybrid of stochastic approximation and the traditional proximal method of multipliers. Under mild conditions, we show that this algorithm exhibits O (K − 1 / 2) expected convergence rates for both objective reduction and constraint violation if parameters in the algorithm are properly chosen, where K denotes the number of iterations. Moreover, we show that, with high probability, the algorithm has a O (log (K) K − 1 / 2) constraint violation bound and O (log 3 / 2 (K) K − 1 / 2) objective bound. Numerical results demonstrate that the proposed algorithm is efficient. History: Accepted byAntonio Frangioni, Area Editor for Design & Analysis ofAlgorithms—Continuous. Funding: This work was supported by the National Natural Science Foundation of China [Grants 11971089, 11731013, 12071055, and 11871135]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplementary Information [https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.1228] or is available from the IJOC GitHub software repository (https://github.com/INFORMSJoC) at http://dx.doi.org/10.5281/zenodo.6818229. [ABSTRACT FROM AUTHOR]

The proximal method of multipliers was proposed by Rockafellar [Augmented Lagrangians and applications of the proximal point algorithm in convex programming. Math Oper Res. 1976;1:97–116] for solving convex programming and it is a kind of proximal point method applied to convex programming. In this paper, we apply this method for solving nonlinear semidefinite programming problems, in which subproblems have better properties than those from the augmented Lagrange method. We prove that, under the linear independence constraint qualification and the strong second-order sufficiency optimality condition, the rate of convergence of the proximal method of multipliers, for a nonlinear semidefinite programming problem, is linear and the ratio constant is proportional to 1/c, where c is the penalty parameter that exceeds a threshold c ∗ > 0. Moreover, the rate of convergence of the proximal method of multipliers is superlinear when the parameter c increases to + ∞. [ABSTRACT FROM AUTHOR]

Providing higher precision Direction of Arrival (DOA) estimation has become a hot topic in the field of array signal processing for parameter estimation in recent years. However, when the physical aperture of the actual array is small, its aperture limitation means that even with super-resolution estimation algorithms, the achievable estimation precision is limited. This paper takes a novel approach by constructing an optimization algorithm using the covariance fitting criterion based on the array output's covariance matrix to fit and obtain the covariance matrix of a large aperture virtual array, thereby providing high-precision angular resolution through virtual aperture expansion. The covariance fitting expansion analysis and discussion are unfolded for both uniform linear arrays (ULAs) and sparse linear arrays (SLAs) under four different scenarios. Theoretical analysis and simulation experiments demonstrate that these methods can enhance the effective performance of angle estimation, especially in low signal-to-noise ratios (SNRs) and at small angular intervals by fitting virtual extended aperture data. [ABSTRACT FROM AUTHOR]