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Flexible Optimization for Cyber-Physical and Human Systems

Andrea Simonetto

TL;DR

Two complementary ways to render convex optimization problems stemming from cyber-physical applications “flexible” are explored, based on robust optimization and convex reformulations and stochastic optimization with decision-dependent distributions.

Abstract

Can we allow humans to pick among different, yet reasonably similar, decisions? Are we able to construct optimization problems whose outcome are sets of feasible, close-to-optimal decisions for human users to pick from, instead of a single, hardly explainable, do-as-I-say ``optimal'' directive? In this paper, we explore two complementary ways to render optimization problems stemming from cyber-physical applications flexible. In doing so, the optimization outcome is a trade off between engineering best and flexibility for the users to decide to do something slightly different. The first method is based on robust optimization and convex reformulations. The second method is stochastic and inspired from stochastic optimization with decision-dependent distributions.

Flexible Optimization for Cyber-Physical and Human Systems

TL;DR

Two complementary ways to render convex optimization problems stemming from cyber-physical applications “flexible” are explored, based on robust optimization and convex reformulations and stochastic optimization with decision-dependent distributions.

Abstract

Can we allow humans to pick among different, yet reasonably similar, decisions? Are we able to construct optimization problems whose outcome are sets of feasible, close-to-optimal decisions for human users to pick from, instead of a single, hardly explainable, do-as-I-say ``optimal'' directive? In this paper, we explore two complementary ways to render optimization problems stemming from cyber-physical applications flexible. In doing so, the optimization outcome is a trade off between engineering best and flexibility for the users to decide to do something slightly different. The first method is based on robust optimization and convex reformulations. The second method is stochastic and inspired from stochastic optimization with decision-dependent distributions.
Paper Structure (14 sections, 2 theorems, 47 equations, 2 figures, 1 table)

This paper contains 14 sections, 2 theorems, 47 equations, 2 figures, 1 table.

Table of Contents

  1. INTRODUCTION
  2. Problem formulation
  3. Deterministic approach
  4. Stochastic approach
  5. SOLVING THE FLEXIBLE PROBLEM
  6. Robust optimization problem
  7. Stochastic optimization problem
  8. A HUMAN-ADAPTED ALGORITHM
  9. Warm-start, guarding, and rounding
  10. NUMERICAL RESULTS
  11. CONCLUSIONS
  12. Derivation of Problem \ref{['eq.ex-rob']}
  13. Proof of Theorem \ref{['th.1']}
  14. Proof of Theorem \ref{['th.2']}

Key Result

Theorem 1

Let $p = [y^\top, \lambda^\top]^\top$. Let Assumption as.1 and the stochastic setting st.setting hold. Assume $\frac{\varepsilon L}{\mu} <1$ and pick the step size $\eta$ as Then the primal-dual method in Eq. pd generates a sequence of points $\{{y}_k, {\lambda}_k\}$, such that in total expectation, with $\varrho := \sqrt{1 - 2 \eta \mu + \eta^2 L^2} + \eta \varepsilon L <1$. $\diamond$

Figures (2)

  • Figure 1: Setting of the problem formulation in two dimensions with the best decision $x^\star$ and the optimal variations around it $\beta^\star$. We give to each user $i$ the optimal set $[x^\star_i - \beta^\star_i, x^\star_i + \beta^\star_i]$.
  • Figure 2: Optimality gap vs. iterations for the considered algorithms. For [B-PD] we indicate mean and standard deviation over $50$ realizations.

Theorems & Definitions (5)

  • Example 1
  • Theorem 1
  • proof
  • Theorem 2
  • proof