# Cardinality Minimization, Constraints, and Regularization: A Survey

@article{Tillmann2021CardinalityMC, title={Cardinality Minimization, Constraints, and Regularization: A Survey}, author={Andreas M. Tillmann and D. Bienstock and A. Lodi and Alexandra Schwartz}, journal={ArXiv}, year={2021}, volume={abs/2106.09606} }

We survey optimization problems that involve the cardinality of variable vectors in constraints or the objective function. We provide a unified viewpoint on the general problem classes and models, and give concrete examples from diverse application fields such as signal and image processing, portfolio selection, or machine learning. The paper discusses general-purpose modeling techniques and broadly applicable as well as problem-specific exact and heuristic solution approaches. While our… Expand

#### References

SHOWING 1-10 OF 374 REFERENCES

Global Optimization for Sparse Solution of Least Squares Problems

- Computer Science
- 2019

A dedicated algorithm is built, based on the homotopy continuation principle, which efficiently computes the relaxed solutions for the three kinds of problems: cardinality-constrained and Cardinality-penalized least-squares, and cardinality minimization under quadratic constraints. Expand

A unified approach to mixed-integer optimization: Nonlinear formulations and scalable algorithms

- Mathematics, Computer Science
- ArXiv
- 2019

This work proposes a unified framework to address a family of classical mixed-integer optimization problems, including network design, facility location, unit commitment, sparse portfolio selection, binary quadratic optimization and sparse learning problems, and establishes that a general-purpose numerical strategy, which combines cutting-plane, first-order and local search methods, solves these problems faster and at a larger scale than state-of-the-art mixed- integer linear or second-order cone methods. Expand

A local relaxation method for the cardinality constrained portfolio optimization problem

- Mathematics, Computer Science
- Comput. Optim. Appl.
- 2012

The proposed local relaxation algorithm explores the inherent structure of the objective function and solves a sequence of small, local, quadratic-programs by first projecting asset returns onto a reduced metric space, followed by clustering in this space to identify sub-groups of assets that best accentuate a suitable measure of similarity amongst different assets. Expand

Sparse learning via Boolean relaxations

- Mathematics, Computer Science
- Math. Program.
- 2015

Novel relaxations for cardinality-constrained learning problems, including least-squares regression as a special but important case, are introduced, and it is shown that randomization based on the relaxed solution offers a principled way to generate provably good feasible solutions. Expand

On handling indicator constraints in mixed integer programming

- Mathematics, Computer Science
- Comput. Optim. Appl.
- 2016

It is argued that aggressive bound tightening is often overlooked in MIP, while it represents a significant building block for enhancing MIP technology when indicator constraints and disjunctive terms are present, and a pair of computationally effective algorithmic approaches are devised that exploit it. Expand

DC formulations and algorithms for sparse optimization problems

- Mathematics, Computer Science
- Math. Program.
- 2018

A DC Algorithm (DCA) is presented, where the dual step at each iteration can be efficiently carried out due to the accessible subgradient of the largest-k norm, and the efficiency of the proposed DCA in comparison with existing methods which have other penalty terms is demonstrated. Expand

Fast Best Subset Selection: Coordinate Descent and Local Combinatorial Optimization Algorithms

- Computer Science, Mathematics
- Oper. Res.
- 2020

This paper empirically demonstrate that a family of L_0-based estimators can outperform the state-of-the-art sparse learning algorithms in terms of a combination of prediction, estimation, and variable selection metrics under various regimes (e.g., different signal strengths, feature correlations, number of samples and features). Expand

Convergence of a Scholtes-type regularization method for cardinality-constrained optimization problems with an application in sparse robust portfolio optimization

- Computer Science, Mathematics
- Comput. Optim. Appl.
- 2018

A Scholtes-type regularization method is applied to obtain a sequence of easier to solve problems and it is shown that such a sequence converges to an S-stationary point, which corresponds to a local minimizer of the original problem under the assumption of convexity. Expand

Convergent Inexact Penalty Decomposition Methods for Cardinality-Constrained Problems

- Computer Science, Mathematics
- J. Optim. Theory Appl.
- 2021

This manuscript considers the problem of minimizing a smooth function with cardinality constraint, and proposes a modified penalty decomposition method, where the exact minimizations with respect to the original variables are replaced by suitable line searches along gradient-related directions. Expand

Mathematical Programs with Cardinality Constraints: Reformulation by Complementarity-Type Conditions and a Regularization Method

- Mathematics, Computer Science
- SIAM J. Optim.
- 2016

This work introduces a mixed-integer formulation whose standard relaxation still has the same solutions as the underlying cardinality-constrained problem; the relation between the local minima is also discussed in detail. Expand