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# 碧桂园政经名家大讲堂系列讲座

## 2019年12月17日 15:20 点击：[_showDynClicks("wbnews", 1129381338, 17129)]

Many practical problems arising in real world applications, mainly in the fields of logistics and telecommunications, can be modeled as unsplittable capacitated network design problem. In this talk, we consider the flow arc-set polyhedron of this problem. By analyzing the property of its nontrivial facet, we design an exact algorithm to solve the associated separation problem such that the constructed inequality guarantees to be facet-defining. To speed up this approach, we present a closed form of the separation problem under some conditions. Furthermore, a new technique is presented to accelerate the exact separation algorithm, which significantly decreases the number of iterations in the algorithm. Finally, a comprehensive computational study on the unsplittable capacitated network design problem is presented to demonstrate the effectiveness of the proposed algorithm.

This talk is about stochastic optimization problems with polynomials. We propose an optimization model with sample averages and perturbations. The Lasserre type Moment-SOS relaxations are used to solve the sample average optimization. Properties of the optimization and its relaxations are studied. Numerical experiments are presented.

We develop a constructive approach to estimating sparse, high-dimensional linear regression models. The approach is a computational algorithm motivated from the KKT conditions for the $\ell_0$-penalized least squares solutions. It generates a sequence of solutions iteratively, based on support detection using primal and dual information and root finding. We refer to the algorithm as SDAR for brevity. Under a sparse Rieze condition on the design matrix and certain other conditions, we show that with high probability, the $\ell_2$ estimation error of the solution sequence decays exponentially to the minimax error bound in $O(\sqrt{J}\log(R))$ steps; and under a mutual coherence condition and certain other conditions, the $\ell_{\infty}$ estimation error decays to the optimal error bound in $O(\log(R))$ steps, where $J$ is the number of important predictors, $R$ is the relative magnitude of the nonzero target coefficients. Computational complexity analysis shows that the cost of SDAR is $O(np)$ per iteration. Moreover the oracle least squares estimator can be exactly recovered with high probability at the same cost if we know the sparsity level. We also consider an adaptive version of SDAR to make it more practical in applications. Numerical comparisons with Lasso, MCP and greedy methods demonstrate that SDAR is competitive with or outperforms them in accuracy and efficiency.

2019年12月17日

 05256881