Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Robust Conic Satisficing

Arjun Ramachandra, Napat Rujeerapaiboon, Melvyn Sim

TL;DR

This paper reframes data-driven optimization under distributional uncertainty as robust conic satisficing, targeting out-of-sample performance via a user-specified risk target $\tau$ and a penalty that measures deviation from nominal data. It provides exact semidefinite formulations for a bi-convex quadratic evaluation with an ellipsoidal uncertainty set and develops safe, tractable approximations via affine dual recourse adaptation for broader conic evaluators, under complete and bounded recourse with polyhedral structures. A novel perspective casting approach ensures the required recourse properties, while affine dual recourse adaptations yield superior computational performance compared to primal counterparts in linear settings. The framework is demonstrated through data-driven portfolio optimization using an exponential utility, where robust satisficing delivers meaningful improvements over empirical and stochastic (Markowitz) approaches, highlighting its practical potential to mitigate overfitting in predictive and prescriptive analytics.

Abstract

In practical optimization problems, we typically model uncertainty as a random variable though its true probability distribution is unobservable to the decision maker. Historical data provides some information of this distribution that we can use to approximately quantify the risk that depends on both the decision and the uncertainty. This empirical optimization approach is vulnerable to the issues of overfitting, which could be overcome by several data-driven robust optimization techniques. To tackle overfitting, Long et.al.(2022) propose a robust satisficing model, which is specified by a performance target and a penalty function that measures the deviation of the uncertainty from its nominal value, and yields solutions with superior out-of-sample performance. We generalize the robust satisficing framework to conic optimization problems with recourse, which has broad applications in predictive and prescriptive analytics. We derive an exact semidefinite optimization formulation for a biconvex quadratic evaluation function, with quadratic penalty and ellipsoidal support set. More importantly, under complete and bounded recourse, and a reasonably chosen polyhedral support set and penalty function, we propose safe approximations that are feasible for any reasonably chosen target. We then demonstrate that the assumption of complete and bounded recourse is not unimpeachable, and then introduce a novel perspective casting technique to derive an equivalent conic optimization problem satisfying the stated assumptions. Computationally, we showcase a study on data-driven portfolio optimization and demonstrate that the robust satisficing solutions can provide significant improvements over the solutions obtained by stochastic optimization models, including the celebrated Markowitz model, which is prone to overfitting.

Robust Conic Satisficing

TL;DR

This paper reframes data-driven optimization under distributional uncertainty as robust conic satisficing, targeting out-of-sample performance via a user-specified risk target and a penalty that measures deviation from nominal data. It provides exact semidefinite formulations for a bi-convex quadratic evaluation with an ellipsoidal uncertainty set and develops safe, tractable approximations via affine dual recourse adaptation for broader conic evaluators, under complete and bounded recourse with polyhedral structures. A novel perspective casting approach ensures the required recourse properties, while affine dual recourse adaptations yield superior computational performance compared to primal counterparts in linear settings. The framework is demonstrated through data-driven portfolio optimization using an exponential utility, where robust satisficing delivers meaningful improvements over empirical and stochastic (Markowitz) approaches, highlighting its practical potential to mitigate overfitting in predictive and prescriptive analytics.

Abstract

In practical optimization problems, we typically model uncertainty as a random variable though its true probability distribution is unobservable to the decision maker. Historical data provides some information of this distribution that we can use to approximately quantify the risk that depends on both the decision and the uncertainty. This empirical optimization approach is vulnerable to the issues of overfitting, which could be overcome by several data-driven robust optimization techniques. To tackle overfitting, Long et.al.(2022) propose a robust satisficing model, which is specified by a performance target and a penalty function that measures the deviation of the uncertainty from its nominal value, and yields solutions with superior out-of-sample performance. We generalize the robust satisficing framework to conic optimization problems with recourse, which has broad applications in predictive and prescriptive analytics. We derive an exact semidefinite optimization formulation for a biconvex quadratic evaluation function, with quadratic penalty and ellipsoidal support set. More importantly, under complete and bounded recourse, and a reasonably chosen polyhedral support set and penalty function, we propose safe approximations that are feasible for any reasonably chosen target. We then demonstrate that the assumption of complete and bounded recourse is not unimpeachable, and then introduce a novel perspective casting technique to derive an equivalent conic optimization problem satisfying the stated assumptions. Computationally, we showcase a study on data-driven portfolio optimization and demonstrate that the robust satisficing solutions can provide significant improvements over the solutions obtained by stochastic optimization models, including the celebrated Markowitz model, which is prone to overfitting.

Paper Structure

This paper contains 18 sections, 19 theorems, 174 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Mitigating ambiguity in data-driven optimization
  3. Empirical and stochastic optimization
  4. Data-driven robust optimization
  5. On classical robust optimization
  6. Data-driven robust satisficing
  7. Tractable formulations and safe approximations
  8. Quadratic evaluation and penalty functions
  9. Conic representable evaluation function
  10. Perspective casting
  11. Polyhedral penalty function and support
  12. Affine dual recourse adaptation
  13. On linear optimization with recourse
  14. Portfolio optimization via expected utility maximization
  15. Acknowledgements:
  16. ...and 3 more sections

Key Result

Theorem 1

fournier2015rate. Let $\mathbb{P}^\Omega$ denote the distribution that governs the distribution of the independent samples $\tilde{\bm{z}}^1,\ldots,\tilde{\bm{z}}^\Omega$ drawn from $\mathbb{P}^{\star}$ for which the empirical distribution $\hat{\mathbb{P}}$ is constructed. When the true data-genera for some function $\varepsilon_q(r):\mathbb{R}_+\rightarrow \mathbb{R}_+$ that decreases to zero at

Figures (4)

  • Figure 1: An illustration of perspective depth on $\mathcal{W} = [\underline w,\bar{w}]$.
  • Figure 2: Robust satisficing objectives obtained from the primal non-affine and the dual affine adaptations.
  • Figure 3: Certainty equivalent comparison between the empirical, stochastic (Markowitz) and robust satisficing investment models.
  • Figure 4: Computational times (in seconds) of the robust optimization (rob opt) and the robust satisficing (rob sat) lot-sizing solutions.

Theorems & Definitions (50)

  • Theorem 1
  • Proposition 1
  • proof
  • Definition 1
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • ...and 40 more