Modified Polyhedral Method for Elicitation of Shape-Free Utility and Conservatism Reduction in Robust Optimization

Abstract

In this paper, we propose a modified polyhedral method to elicit a decision maker's (DM's) nonlinear univariate utility function, which does not rely on explicit information about the shape structure, Lipschitz modulus, and the inflection point of the utility. The method is inspired by Toubia et al. (2004) for elicitation of the linear multi-variate utility and the success of the modification needs to overcome two main difficulties. First, we use the continuous piecewise linear function (PLF) to approximate the nonlinear utility and represent the PLF in terms of the vector of increments of linear pieces. Subsequently, elicitation of the nonlinear utility corresponds to reducing the polyhedral feasible set of the vectors of increments. Second, we reduce the size of the polyhedron by successive hyperplane cuts constructed by adaptively generating new queries (pairwise comparison lotteries) where the parameters of the lotteries are obtained by solving some optimization problems. In this reduction procedure, direction error of the cut hyperplane may occur due to the PLF approximation error. To tackle the issue, we develop a strategy by adding the support points of new lotteries to the set of breakpoints of the PLF. As an application, we use all the responses to the queries to construct an ambiguity set of utility functions which allows one to make decisions based on the worst-case utility and apply the modified polyhedral method in a preference robust optimization problem with proper conservatism reduction scheme. The preliminary numerical test results show that the proposed methods work very well.

Figures (12)

• Figure 1: (a) PLF of normalized utility $u^*(x)=1-e^{-10x}$ over $[-0.5,0]$, ${\cal X}=\{x_1,\cdots,x_5\}$, $x_0=0.8*x_4+0.2*x_5$. $\phi_i(x_0)=0$, for $i=1,2,3$, and $\phi_4(x_0)=\frac{x_0-x_4}{x_5-x_4}=0.2$, $\mathds{1}_{(x_i,x_{i+1}]}(x_0)=0$, for $i=1,2,3$ and $\mathds{1}_{(x_4,x_{5}]}(x_0)=1$, ${\bm g}(x_0)=(\mathds{1}_{(x_4,x_5]}(x_0),\mathds{1}_{(x_4,x_5]}(x_0)$, $\mathds{1}_{(x_4,x_5]}(x_0),\phi_4(x_0)) =(1,1,1,0.2)$, $u_N(x_0)=(v_1,v_2,v_3,v_4)^\top (1,1,1,0.2)=\sum_{i=1}^3 v_i+0.2v_4$. (b) The red segments jointing three points ${\bm g}(x_1)=(0,0)$, ${\bm g}(x_2)=(1,0)$, ${\bm g}(x_3)=(1,1)$ represents the range of ${\bm g}(\cdot)\in {\rm I\!R}^2$ defined as in (\ref{['eq:g(t)-PRO']}). (c) The red segments jointing four points ${\bm g}(x_1)=(0,0,0)$, ${\bm g}(x_2)=(1,0,0)$, ${\bm g}(x_3)=(1,1,0)$, ${\bm g}(x_4)=(1,1,1)$, represents the range of vector-valued function ${\bm g}(\cdot)\in {\rm I\!R}^3$.
• Figure 2: (a) Illustration of $\Upsilon_{\bm A}$, $\Upsilon_{\bm B}$, $\Upsilon_{\bm A}^{=}$ and $\Upsilon_{\bm B}^{=}$. The gray area represents $\{{\bm d}\in {\rm I\!R}^2:({\bm R}{\bm c}^1)^\top {\bm d}\leq D\}$, the red area represents $\Upsilon_{\bm A}$ defined as in (\ref{['eq:Upsilon_A']}), the red segments represents $\Upsilon_{\bm B}$ defined as in (\ref{['eq:Upsilon_B']}). The blue segment represents $\Upsilon_{\bm A}^=$ and the green diamond point represents $\Upsilon_{\bm B}^=$ given in (\ref{['eq:UpsilonAB']}). (b) Illustration of feasible sets and objective functions of problems (\ref{['eq:G_A_G_B-rfmlt']}). The triangle area is the intersection of $\{{\bm d}\in {\rm I\!R}^2:({\bm R}{\bm c}^1)^\top {\bm d}\leq D\}$ and $\Upsilon_{\bm A}$, which is the feasible set $\{{\bm d}\in {\rm I\!R}^2: ({\bm R}{\bm c}^1)^\top {\bm d}\leq D,{\bm d}\in \Upsilon_{\bm A}\}$ of problem in (\ref{['eq:G_A_G_B-rfmlt-b']}). The red segment on $d_1$-axis represents the intersection of $\{{\bm d}\in {\rm I\!R}^2:({\bm R}{\bm c}^1)^\top {\bm d}\leq D\}$ and $\Upsilon_{\bm B}$, which is the feasible set $\{{\bm d}\in {\rm I\!R}^2: ({\bm R}{\bm c}^1)^\top {\bm d}\leq D, {\bm d}\in \Upsilon_{\bm B}\}$ of problem in (\ref{['eq:G_A_G_B-rfmlt-a']}). The dashed blue and green lines represent the contours of the functions $l_2({\bm d}) := ({\bm R}{\bm v}_2^1)^\top {\bm d}$ and $l_1({\bm d}) := ({\bm R}{\bm v}_1^1)^\top {\bm d}$ over ${\rm I\!R}^2$. (c) Illustration of the cut direction $G_{\bm A}(r_1^1,r_3^1,p^1)-G_{\bm B}(r_2^1)$ for the $1$-st cut hyperplane. Problems (\ref{['eq:G_A_G_B-rfmlt']}) are indeed the maximization of $l_2({\bm d})$ over the feasible set $\{{\bm d}\in {\rm I\!R}^2: ({\bm R}{\bm c}^1)^\top {\bm d}\leq D, {\bm d}\in \Upsilon_{\bm A}\}$ with the maximum being attained at the blue diamond point ${\bm d}_{\bm A}^*=G_{\bm A}(r_1^1,r_3^1,p^1)$, and the maximization of $l_1({\bm d})$ over $\{{\bm d}\in {\rm I\!R}^2:({\bm R}{\bm c}^1)^\top {\bm d}\leq D, {\bm d}\in \Upsilon_{\bm B}\}$ with the maximum being attained at the green diamond point ${\bm d}_{\bm B}^*=G_{\bm B}(r_2^1)$.
• Figure 3: (a) Illustration of the set $\Upsilon_{\bm A}\in {\rm I\!R}^3$, $\Upsilon_{\bm B}\in {\rm I\!R}^3$, $\Upsilon_{\bm A}^=$, and $\Upsilon_{\bm B}^=$, . (b) Illustration of feasible sets $\{{\bm d}\in {\rm I\!R}^3: ({\bm R}{\bm c}^1)^\top {\bm d}\leq D, {\bm d}\in \Upsilon_{\bm A}\}$ and $\{{\bm d}\in {\rm I\!R}^3: ({\bm R}{\bm c}^1)^\top {\bm d}\leq D, {\bm d}\in \Upsilon_{\bm A}\}$, and the objective functions $P_1({\bm d})$ and $P_2({\bm d})$ of problems in (\ref{['eq:G_A_G_B-rfmlt']}). (c) The maximization of functions $P_1({\bm d})$ and $P_2({\bm d})$ is achieved at ${\bm d}_{\bm A}^*=G_{\bm A}(r_1^m,r_3^m,p^m)$ and ${\bm d}_{\bm B}^*=G_{\bm B}(r_2^m)$ respectively. Illustration of cut direction $G_{\bm A}(r_1^m,r_3^m,p^m)-G_{\bm B}(r_2^m)$.
• Figure 4: Visualization of the cut hyperplanes and the resulting polyhedron ${\cal V}_4^m$ for $m=1,2$. (a) In $d_1Od_2$-plane, the grey triangle with vertices $A$, $B$ and $C$ represents $\{({\bm d},0)\in {\rm I\!R}^3: {\bm d}\in {\cal V}_4^1\}$ and the one framed by the red dashed lines with vertices $A^0$, $B^0$ and $C^0$ is $\{{\bm d}\in {\rm I\!R}^3: {\bm d}={\bm R}{\bm v},{\bm v}\in {\cal V}_4^1\}$ after the first cut hyperplane. The red point is the vector of increments $(v_1^*,v_2^*,v_3^*)=(0.18,0.81,0.01)$ with $v_i^*=\frac{u^*(x_{i+1})-u^*(x_i)}{u^*(\bar{x})-u^*(\underline{x})}$, for $i=1,2,3$, where $u^*$ is the true utility. (b) shows the 2nd cut and the resulting polyhedrons $\{({\bm v},0)\in {\rm I\!R}^3: {\bm v}\in {\cal V}_4^2\}$ enclosed by the red dotted lines with vertices $D$, $E$, $F$ and $G$ and the one with vertices $D^0$, $E^0$, $F^0$ and $G^0$ is the set $\{{\bm d}\in {\rm I\!R}^3:{\bm d}={\bm R}{\bm v},{\bm v}\in {\cal V}_4^2\}$.
• Figure 5: The dotted curves in (a) and (b) corresponds to the PLA utility functions constructed with analytic centers ${\bm c}^{M}$, the black curve represents the PLA $u_N^*$. The green area specifies the range $[\underline{u}_i,\bar{u}_i]$ for $i\in [N]$. (c) The blue curves depict the change of CPU time (seconds) as the number of queries $M$ increases. The black curves shows changes of the number of breakpoints ($N_M=|{\cal X}_M|$) as $M$ varies.

Theorems & Definitions (25)

• Definition 2.1: Increment-Based PLFs, HZXZ22
• Proposition 2.1: Characterization of ${\bm g}(x)$, HZXZ22
• Proposition 2.2: Linear Representation of ${\bm g}(x)$, HZXZ22
• Remark 2.1: PLA Utility $u_N(x)$ (\ref{['eq:PLA_utility']}) v.s. Linear Utility $U_{\bm x}$ (\ref{['eq:Ux-market']})
• Definition 3.1: Pairwise Comparison of Lotteries under EUT
• Proposition 3.1: Characterization of Queries in Terms of Vectors of Increments
• Definition 3.2: Iterative Construction of the Polyhedrons
• Definition 3.3: Analytic Center
• Definition 3.4: Sonnevend's Inner Ellipsoid Son85
• Definition 4.1: Construction of ($r_1^m$, $r_2^m$, $r_3^m$, $p^m$)