Niharika Challapalli

Niharika Challapalli, Masaaki Nagahara, Mathukumalli Vidyasagar

In this paper, we consider hands-off control via minimization of the CLOT (Combined $L$-One and Two) norm. The maximum hands-off control is the $L^0$-optimal (or the sparsest) control among all feasible controls that are bounded by a specified value and transfer the state from a given initial state to the origin within a fixed time duration. In general, the maximum hands-off control is a bang-off-bang control taking values of $\pm 1$ and $0$. For many real applications, such discontinuity in the control is not desirable. To obtain a continuous but still relatively sparse control, we propose to use the CLOT norm, a convex combination of $L^1$ and $L^2$ norms. We show by numerical simulations that the CLOT control is continuous and much sparser (i.e. has longer time duration on which the control takes 0) than the conventional EN (elastic net) control, which is a convex combination of $L^1$ and squared $L^2$ norms. We also prove that the CLOT control is continuous in the sense that, if $O(h)$ denotes the sampling period, then the difference between successive values of the CLOT-optimal control is $O(\sqrt{h})$, which is a form of continuity. Also, the CLOT formulation is extended to encompass constraints on the state variable.

Niharika Challapalli, Masaaki Nagahara, Mathukumalli Vidyasagar

In this paper, we consider hands-off control via minimization of the CLOT (Combined L-One and Two) norm. The maximum hands-off control is the L0-optimal (or the sparsest) control among all feasible controls that are bounded by a specified value and transfer the state from a given initial state to the origin within a fixed time duration. In general, the maximum hands-off control is a bang-off-bang control taking values of +1, -1, and 0. For many real applications, such discontinuity in the control is not desirable. To obtain a continuous but still relatively sparse control, we propose to use the CLOT norm, a convex combination of L1 and L2 norms. We show by numerical simulation that the CLOT control is continuous and much sparser (i.e. has longer time duration on which the control takes 0) than the conventional EN (elastic net) control, which is a convex combination of L1 and squared L2 norms.

Mehmet Eren Ahsen, Niharika Challapalli, Mathukumalli Vidyasagar

In this paper we introduce a new optimization formulation for sparse regression and compressed sensing, called CLOT (Combined L-One and Two), wherein the regularizer is a convex combination of the $\ell_1$- and $\ell_2$-norms. This formulation differs from the Elastic Net (EN) formulation, in which the regularizer is a convex combination of the $\ell_1$- and $\ell_2$-norm squared. It is shown that, in the context of compressed sensing, the EN formulation does not achieve robust recovery of sparse vectors, whereas the new CLOT formulation achieves robust recovery. Also, like EN but unlike LASSO, the CLOT formulation achieves the grouping effect, wherein coefficients of highly correlated columns of the measurement (or design) matrix are assigned roughly comparable values. It is already known LASSO does not have the grouping effect. Therefore the CLOT formulation combines the best features of both LASSO (robust sparse recovery) and EN (grouping effect). The CLOT formulation is a special case of another one called SGL (Sparse Group LASSO) which was introduced into the literature previously, but without any analysis of either the grouping effect or robust sparse recovery. It is shown here that SGL achieves robust sparse recovery, and also achieves a version of the grouping effect in that coefficients of highly correlated columns belonging to the same group of the measurement (or design) matrix are assigned roughly comparable values.