27 research outputs found
Inverse Optimization: Closed-form Solutions, Geometry and Goodness of fit
In classical inverse linear optimization, one assumes a given solution is a
candidate to be optimal. Real data is imperfect and noisy, so there is no
guarantee this assumption is satisfied. Inspired by regression, this paper
presents a unified framework for cost function estimation in linear
optimization comprising a general inverse optimization model and a
corresponding goodness-of-fit metric. Although our inverse optimization model
is nonconvex, we derive a closed-form solution and present the geometric
intuition. Our goodness-of-fit metric, , the coefficient of
complementarity, has similar properties to from regression and is
quasiconvex in the input data, leading to an intuitive geometric
interpretation. While is computable in polynomial-time, we derive a
lower bound that possesses the same properties, is tight for several important
model variations, and is even easier to compute. We demonstrate the application
of our framework for model estimation and evaluation in production planning and
cancer therapy
Algorithms for Inverse Optimization Problems
We study inverse optimization problems, wherein the goal is to map given solutions to an underlying optimization problem to a cost vector for which the given solutions are the (unique) optimal solutions. Inverse optimization problems find diverse applications and have been widely studied. A prominent problem in this field is the inverse shortest path (ISP) problem [D. Burton and Ph.L. Toint, 1992; W. Ben-Ameur and E. Gourdin, 2004; A. Bley, 2007], which finds applications in shortest-path routing protocols used in telecommunications. Here we seek a cost vector that is positive, integral, induces a set of given paths as the unique shortest paths, and has minimum l_infty norm. Despite being extensively studied, very few algorithmic results are known for inverse optimization problems involving integrality constraints on the desired cost vector whose norm has to be minimized.
Motivated by ISP, we initiate a systematic study of such integral inverse optimization problems from the perspective of designing polynomial time approximation algorithms. For ISP, our main result is an additive 1-approximation algorithm for multicommodity ISP with node-disjoint commodities, which we show is tight assuming P!=NP. We then consider the integral-cost inverse versions of various other fundamental combinatorial optimization problems, including min-cost flow, max/min-cost bipartite matching, and max/min-cost basis in a matroid, and obtain tight or nearly-tight approximation guarantees for these. Our guarantees for the first two problems are based on results for a broad generalization, namely integral inverse polyhedral optimization, for which we also give approximation guarantees. Our techniques also give similar results for variants, including l_p-norm minimization of the integral cost vector, and distance-minimization from an initial cost vector
Data-Driven Estimation in Equilibrium Using Inverse Optimization
Equilibrium modeling is common in a variety of fields such as game theory and
transportation science. The inputs for these models, however, are often
difficult to estimate, while their outputs, i.e., the equilibria they are meant
to describe, are often directly observable. By combining ideas from inverse
optimization with the theory of variational inequalities, we develop an
efficient, data-driven technique for estimating the parameters of these models
from observed equilibria. We use this technique to estimate the utility
functions of players in a game from their observed actions and to estimate the
congestion function on a road network from traffic count data. A distinguishing
feature of our approach is that it supports both parametric and
\emph{nonparametric} estimation by leveraging ideas from statistical learning
(kernel methods and regularization operators). In computational experiments
involving Nash and Wardrop equilibria in a nonparametric setting, we find that
a) we effectively estimate the unknown demand or congestion function,
respectively, and b) our proposed regularization technique substantially
improves the out-of-sample performance of our estimators.Comment: 36 pages, 5 figures Additional theorems for generalization guarantees
and statistical analysis adde
Inverse Optimization of Convex Risk Functions
The theory of convex risk functions has now been well established as the
basis for identifying the families of risk functions that should be used in
risk averse optimization problems. Despite its theoretical appeal, the
implementation of a convex risk function remains difficult, as there is little
guidance regarding how a convex risk function should be chosen so that it also
well represents one's own risk preferences. In this paper, we address this
issue through the lens of inverse optimization. Specifically, given solution
data from some (forward) risk-averse optimization problems we develop an
inverse optimization framework that generates a risk function that renders the
solutions optimal for the forward problems. The framework incorporates the
well-known properties of convex risk functions, namely, monotonicity,
convexity, translation invariance, and law invariance, as the general
information about candidate risk functions, and also the feedbacks from
individuals, which include an initial estimate of the risk function and
pairwise comparisons among random losses, as the more specific information. Our
framework is particularly novel in that unlike classical inverse optimization,
no parametric assumption is made about the risk function, i.e. it is
non-parametric. We show how the resulting inverse optimization problems can be
reformulated as convex programs and are polynomially solvable if the
corresponding forward problems are polynomially solvable. We illustrate the
imputed risk functions in a portfolio selection problem and demonstrate their
practical value using real-life data
Inverse Optimization with Noisy Data
Inverse optimization refers to the inference of unknown parameters of an
optimization problem based on knowledge of its optimal solutions. This paper
considers inverse optimization in the setting where measurements of the optimal
solutions of a convex optimization problem are corrupted by noise. We first
provide a formulation for inverse optimization and prove it to be NP-hard. In
contrast to existing methods, we show that the parameter estimates produced by
our formulation are statistically consistent. Our approach involves combining a
new duality-based reformulation for bilevel programs with a regularization
scheme that smooths discontinuities in the formulation. Using epi-convergence
theory, we show the regularization parameter can be adjusted to approximate the
original inverse optimization problem to arbitrary accuracy, which we use to
prove our consistency results. Next, we propose two solution algorithms based
on our duality-based formulation. The first is an enumeration algorithm that is
applicable to settings where the dimensionality of the parameter space is
modest, and the second is a semiparametric approach that combines nonparametric
statistics with a modified version of our formulation. These numerical
algorithms are shown to maintain the statistical consistency of the underlying
formulation. Lastly, using both synthetic and real data, we demonstrate that
our approach performs competitively when compared with existing heuristics
Data-driven Inverse Optimization with Imperfect Information
In data-driven inverse optimization an observer aims to learn the preferences
of an agent who solves a parametric optimization problem depending on an
exogenous signal. Thus, the observer seeks the agent's objective function that
best explains a historical sequence of signals and corresponding optimal
actions. We focus here on situations where the observer has imperfect
information, that is, where the agent's true objective function is not
contained in the search space of candidate objectives, where the agent suffers
from bounded rationality or implementation errors, or where the observed
signal-response pairs are corrupted by measurement noise. We formalize this
inverse optimization problem as a distributionally robust program minimizing
the worst-case risk that the {\em predicted} decision ({\em i.e.}, the decision
implied by a particular candidate objective) differs from the agent's {\em
actual} response to a random signal. We show that our framework offers rigorous
out-of-sample guarantees for different loss functions used to measure
prediction errors and that the emerging inverse optimization problems can be
exactly reformulated as (or safely approximated by) tractable convex programs
when a new suboptimality loss function is used. We show through extensive
numerical tests that the proposed distributionally robust approach to inverse
optimization attains often better out-of-sample performance than the
state-of-the-art approaches