A parameter-free unconstrained reformulation for nonsmooth problems with convex constraints

In the present paper we propose to rewrite a nonsmooth problem subjected to convex constraints as an unconstrained problem. We show that this novel formulation shares the same global and local minima with the original constrained problem. Moreover, the reformulation can be solved with standard nonsmooth optimization methods if we are able to make projections onto the feasible sets. Numerical evidence shows that the proposed formulation compares favorably against state-of-art approaches. Code can be found at https://github.com/jth3galv/dfppm.

Parameter Optimization of Droop Controllers for Microgrids in Islanded Mode by the SQP Method with Gradient Sampling

For enhancing the stability of the microgrid operation, this paper proposes an optimization model considering the small-signal stability constraint. Due to the nonsmooth property of the spectral abscissa function, the droop controller parameters’ optimization is a nonsmooth optimization problem. The Sequential Quadratic Programming with Gradient Sampling (SQP-GS) is implemented to optimize the droop controller parameters for solving the nonsmooth problem. The SQP-GS method can guarantee the solution of the optimization problem globally and efficiently converges to stationary points with probability of one. In the current iteration, the gradient of the nonsmooth function can be evaluated on a set of randomly generated nearby points by computing closed-form sensitivities. A test on the microgrid system shows that the optimality and the efficiency of the SQP-GS are better than those of the heuristic algorithms.

Avoiding Optimal Mean ℓ2,1-Norm Maximization-Based Robust PCA for Reconstruction

Robust principal component analysis (PCA) is one of the most important dimension-reduction techniques for handling high-dimensional data with outliers. However, most of the existing robust PCA presupposes that the mean of the data is zero and incorrectly utilizes the average of data as the optimal mean of robust PCA. In fact, this assumption holds only for the squared [Formula: see text]-norm-based traditional PCA. In this letter, we equivalently reformulate the objective of conventional PCA and learn the optimal projection directions by maximizing the sum of projected difference between each pair of instances based on [Formula: see text]-norm. The proposed method is robust to outliers and also invariant to rotation. More important, the reformulated objective not only automatically avoids the calculation of optimal mean and makes the assumption of centered data unnecessary, but also theoretically connects to the minimization of reconstruction error. To solve the proposed nonsmooth problem, we exploit an efficient optimization algorithm to soften the contributions from outliers by reweighting each data point iteratively. We theoretically analyze the convergence and computational complexity of the proposed algorithm. Extensive experimental results on several benchmark data sets illustrate the effectiveness and superiority of the proposed method.

Nonsmooth Recursive Identification of Sandwich Systems with Backlash-Like Hysteresis

A recursive gradient identification algorithm based on the bundle method for sandwich systems with backlash-like hysteresis is presented in this paper. In this method, a dynamic parameter estimation scheme based on a subgradient is developed to handle the nonsmooth problem caused by the backlash embedded in the system. The search direction of the algorithm is estimated based on the so-called bundle method. Then, the convergence of the algorithm is discussed. After that, simulation results on a nonsmooth sandwich system are presented to validate the proposed estimation algorithm. Finally, the application of the proposed method to anX-Ymoving positioning stage is illustrated.

Constructive Filtering Algorithms for Delayed Systems With Uncertain Statistics

A stochastic optimal guaranteed estimation problem for dynamic delayed systems with uncertain statistics is considered. The solution of this problem reduces to a complex nonsmooth extremal problem. To obtain an approximate solution, the nonsmooth problem is replaced by a smooth one. Constructive filtering algorithms are obtained from an approximate solution of the smooth problem under the assumption that the delay is small in comparison with the observation time. Estimates for the nonoptimality levels of the proposed filtering algorithms are derived.

A note on nondifferentiable symmetric duality

Under suitable hypotheses on the function f, the two constrained minimization problems:are well known each to be dual to the other. This symmetric duality result is now extended to a class of nonsmooth problems, assuming some convexity hypotheses. The first problem is generalized to:in which T and S are convex cones, S* is the dual cone of S, and ∂y denotes the subdifferential with respect to y. The usual method of proof uses second derivatives, which are no longer available. Therefore a different method is used, where a nonsmooth problem is approximated by a sequence of smooth problems. This duality result confirms a conjecture by Chandra, which had previously been proved only in special cases.