## Abstract

Recent exploration of optimal individualized decision rules (IDRs) for patients in precision medicine has attracted a lot of attention due to the heterogeneous responses of patients to different treatments. In the existing literature of precision medicine, an optimal IDR is defined as a decision function mapping from the patients' covariate space into the treatment space that maximizes the expected outcome of each individual. Motivated by the concept of optimized certainty equivalent (OCE) introduced originally in [Ben-Tal and Teboulle, Manag. Sci., 32 (1986), pp. 1445-1466] that includes the popular conditional-value-of risk (CVaR) [Rockafellar and Uryasev, J. Risk, 2 (2000), pp. 21-42], we propose a decision-rule-based optimized covariates dependent equivalent (CDE) for individualized decision-making problems. Our proposed IDR-CDE broadens the existing expected-mean outcome framework in precision medicine and enriches the previous concept of the OCE. Under a functional margin description of the decision rule modeled by an indicator function as in the literature of large-margin classifiers, we study the mathematical problem of estimating an optimal IDR in two cases: In one case, an optimal solution can be obtained “explicitly” that involves the implicit evaluation of an OCE; the other case requires the numerical solution of an empirical minimization problem obtained by sampling the underlying distributions of the random variables involved. A major challenge of the latter optimization problem is that it involves a discontinuous objective function. We show that, under a mild condition at the population level of the model, the epigraphical formulation of this empirical optimization problem is a piecewise affine (thus difference-of-convex (dc)) constrained dc (thus nonconvex) program. A simplified dc algorithm is employed to solve the resulting dc program whose convergence to a new kind of stationary solution is established. Numerical experiments demonstrate that our overall approach outperforms existing methods in estimating optimal IDRs under heavy-tail distributions of the data. In addition to providing a risk-based approach for individualized medical treatments, which is new in the area of precision medicine, the main contributions of this work in general include the broadening of the concept of the OCE, the epigraphical description of the empirical IDR-CDE minimization problem and its equivalent dc formulation, and the optimization of the resulting piecewise affine constrained dc program.

Original language | English (US) |
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Pages (from-to) | 2337-2362 |

Number of pages | 26 |

Journal | SIAM Journal on Optimization |

Volume | 29 |

Issue number | 3 |

DOIs | |

State | Published - 2019 |

Externally published | Yes |

### Bibliographical note

Funding Information:The work of the first and third authors was partially supported by the National Science Foundation grant IIS-1632951 and the National Institute of Health grant R01GM126550. The work of the second and fourth authors was partially supported by the National Science Foundation grant IIS-1632971.

Publisher Copyright:

© 2019 Society for Industrial and Applied Mathematics

## Keywords

- Conditional value-at-risk
- Dc programming for discontinuous optimization
- Individualized decision making
- Optimized covariate-dependent equivalent
- Precision medicine