Sparsity Constrained Nonlinear Optimization: Optimality Conditions and Algorithms (1203.4580v1)

Abstract: This paper treats the problem of minimizing a general continuously differentiable function subject to sparsity constraints. We present and analyze several different optimality criteria which are based on the notions of stationarity and coordinate-wise optimality. These conditions are then used to derive three numerical algorithms aimed at finding points satisfying the resulting optimality criteria: the iterative hard thresholding method and the greedy and partial sparse-simplex methods. The first algorithm is essentially a gradient projection method while the remaining two algorithms are of coordinate descent type. The theoretical convergence of these methods and their relations to the derived optimality conditions are studied. The algorithms and results are illustrated by several numerical examples.

• The paper introduces necessary optimality conditions—basic feasibility, L-stationarity, and coordinate-wise optimality—to structure sparsity constrained problems.
• It develops iterative algorithms such as the Iterative Hard Thresholding and sparse-simplex methods to achieve L-stationarity and boost computational efficiency.
• The methods offer practical applications in compressed sensing, image processing, and statistical model selection, opening avenues for future research.

Sparsity Constrained Nonlinear Optimization: An Overview

The paper of sparsity constrained nonlinear optimization has garnered significant interest given its extensive applicability across various fields including signal processing, applied mathematics, and statistics. The focal point of this paper by Amir Beck and Yonina C. Eldar is the minimization of continuously differentiable functions under sparsity constraints, where the primary challenge lies in the nonconvexity and discontinuity of such constraints. This paper explores necessary optimality conditions alongside developing algorithms aimed at finding solutions that meet these conditions.

Optimality Conditions

The paper introduces and scrutinizes three categories of necessary optimality conditions: basic feasibility (BF), L-stationarity, and coordinate-wise (CW) optimality. The BF condition, which serves as the weakest of the three, requires the gradient to be zero over the support set of a sparsity-constrained variable. Despite its simplicity, the BF condition often falls short in practice due to its lack of stringency.

L-stationarity is framed as a stronger condition, necessitated by the Lipschitz continuity of the function’s gradient. It provides a more restrictive criterion, especially when the parameter $L$ is carefully selected. The authors underscore that L-stationarity encapsulates the notion of a coordinate descent method, where a point is considered stationary if it is an orthogonal projection onto the feasible set parameterized by $L$.

CW-minima present the strongest of the conditions. It ensures that the optimality is reached by independently minimizing the function with respect to each variable while keeping others fixed. The authors demonstrate that CW-minima not only imply L-stationarity but often encompass a broader class of higher quality solutions.

Numerical Algorithms

The authors introduce the Iterative Hard Thresholding (IHT) method, a fixed point iteration approach aimed at achieving L-stationarity under the assumption of Lipschitz continuity. This method is particularly highlighted for its applicability within the constraints specified by $L > L(f)$. The greedy sparse-simplex method is another key algorithm explored, promoting a coordinate descent strategy that optimizes the function over a potentially nonconvex feasible set without a need for gradient Lipschitz continuity.

A variation termed the partial sparse-simplex method is also discussed, with a simplified index selection strategy that makes it computationally more efficient than its greedy counterpart. The authors emphasize the practical advantage of this method, particularly when the Lipschitz constant is unknown.

Implications and Future Directions

The implications of this research are significant for fields where sparsity is essential, including compressed sensing, image processing, and statistical model selection. The derivation of optimality conditions and the associated algorithms provide a robust foundation for solving complex sparsity constrained optimization problems.

This paper also opens avenues for further exploration into more sophisticated sparsity constraints that may involve additional parameters or dual sparsity notions. The exploration into improved algorithmic formulations, perhaps employing machine learning principles, could further enhance solution robustness and computational efficiency.

Conclusion

Amir Beck and Yonina C. Eldar have provided a comprehensive investigation into sparsity constrained nonlinear optimization, presenting optimality conditions and algorithms that advance the understanding and potential applications of this mathematical problem. By systematically addressing the difficulties posed by nonconvex and discontinuous constraints, this paper contributes significantly to the field, laying groundwork for both theoretical advancements and practical implementations. Future research, catalyzed by these findings, could further expand the applicability and efficiency of sparsity-based optimization techniques across various scientific domains.