Optimization with Discrete Simultaneous Perturbation Stochastic Approximation Using Noisy Loss Function Measurements

Discrete stochastic optimization considers the problem of minimizing (or maximizing) loss functions defined on discrete sets, where only noisy measurements of the loss functions are available. The discrete stochastic optimization problem is widely applicable in practice, and many algorithms have been considered to solve this kind of optimization problem. Motivated by the efficient algorithm of simultaneous perturbation stochastic approximation (SPSA) for continuous stochastic optimization problems, we introduce the middle point discrete simultaneous perturbation stochastic approximation (DSPSA) algorithm for the stochastic optimization of a loss function defined on a p-dimensional grid of points in Euclidean space. We show that the sequence generated by DSPSA converges to the optimal point under some conditions. Consistent with other stochastic approximation methods, DSPSA formally accommodates noisy measurements of the loss function. We also show the rate of convergence analysis of DSPSA by solving an upper bound of the mean squared error of the generated sequence. In order to compare the performance of DSPSA with the other algorithms such as the stochastic ruler algorithm (SR) and the stochastic comparison algorithm (SC), we set up a bridge between DSPSA and the other two algorithms by comparing the probability in a big-O sense of not achieving the optimal solution. We show the theoretical and numerical comparison results of DSPSA, SR, and SC. In addition, we consider an application of DSPSA towards developing optimal public health strategies for containing the spread of influenza given limited societal resources

Parallel Deterministic and Stochastic Global Minimization of Functions with Very Many Minima

The optimization of three problems with high dimensionality and many local minima are investigated under five different optimization algorithms: DIRECT, simulated annealing, Spall’s SPSA algorithm, the KNITRO package, and QNSTOP, a new algorithm developed at Indiana University

Calibration of Aperture Arrays in Time Domain Using the Simultaneous Perturbation Algorithm

Online calibration is desired in antenna arrays of ultrawide bandwidth. This study proposes a time domain calibration method based on the simultaneous perturbation algorithm. Two objective functions were established: power of the received signal at array output; or combination of power and correlation coefficient between the signal at array output and a target signal. For both criteria, the convergence settings require only two measurements at each iteration. One advantage of the method is that the entire signal operation for calibration is performed in the time domain. This is achieved by resolving the effects of distortion on time delay of each channel, which accounts for both amplitude and phase distortions at different frequencies. Therefore, the proposed method significantly increased the calibration efficiency for ultra-wideband antenna arrays. Since time delay coefficients for calibration associated with array elements were determined independently due to characteristic of the simultaneous perturbation, estimation accuracy of the method is tangential to the number of elements in the array, and is mainly dependent on the convergence conditions. This gives the method an additional distinct advantage for calibrating large-scale antenna arrays with ultrawide bandwidth. An estimation accuracy of 99% on time delay adjustments has been achieved and demonstrated

Queueing analysis of opportunistic scheduling with spatially correlated channels

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