Optimal experimental design (OED) is the general formalism of sensor placement and decisions about the data collection strategy for engineered or natural experiments. This approach is prevalent in many critical fields such as battery design, numerical weather prediction, geosciences, and environmental and urban studies. State-of-the-art computational methods for experimental design, however, do not accommodate correlation structure in observational errors produced by many expensive-to-operate devices such as X-ray machines or radar and satellite retrievals. Discarding evident data correlations leads to biased results, poor data collection decisions, and waste of valuable resources. We present a general formulation of the OED formalism for model-constrained large-scale Bayesian linear inverse problems, where measurement errors are generally correlated. The proposed approach utilizes the Hadamard product of matrices to formulate the weighted likelihood and is valid for both finite- and infinite- dimensional Bayesian inverse problems. We also discuss widely used approaches for relaxation of the binary OED problem, in light of the proposed pointwise weighting approach, and present a clear interpretation of the relaxed design and its effect on the observational error covariance. Extensive numerical experiments are carried out for empirical verification of the proposed approach by using an advection-diffusion model, where the objective is to optimally place a small set of sensors, under a limited budget, to predict the concentration of a contaminant in a bounded domain.
We present a method for computing A-optimal sensor placements for infinite-dimensional Bayesian linear inverse problems governed by PDEs with irreducible model uncertainties. Here, irreducible uncertainties refers to uncertainties in the model that exist in addition to the parameters in the inverse problem, and that cannot be reduced through observations. Specifically, given a statistical distribution for the model uncertainties, we compute the optimal design that minimizes the expected value of the posterior covariance trace. The expected value is discretized using Monte Carlo leading to an o…
We introduce optimal transport based resampling in adaptive SMC. We consider elliptic inverse problems of inferring hydraulic conductivity from pressure measurements. We consider two parametrizations of hydraulic conductivity: by Gaussian random field, and by a set of scalar (non-)Gaussian distributed parameters and Gaussian random fields. We show that for scalar parameters optimal transport based SMC performs comparably to monomial based SMC but for Gaussian high-dimensional random fields optimal transport based SMC outperforms monomial based SMC. When comparing to ensemble Kalman inversion w…
We consider optimal design of infinite-dimensional Bayesian linear inverse problems governed by partial differential equations that contain secondary reducible model uncertainties, in addition to the uncertainty in the inversion parameters. By reducible uncertainties we refer to parametric uncertainties that can be reduced through parameter inference. We seek experimental designs that minimize the posterior uncertainty in the primary parameters, while accounting for the uncertainty in secondary parameters. We accomplish this by deriving a marginalized A-optimality criterion and developing an e…
In this paper, we consider the inverse optimal control problem for the discrete-time linear quadratic regulator, over finite-time horizons. Given observations of the optimal trajectories, and optimal control inputs, to a linear time-invariant system, the goal is to infer the parameters that define the quadratic cost function. The well-posedness of the inverse optimal control problem is first justified. In the noiseless case, when these observations are exact, we analyze the identifiability of the problem and provide sufficient conditions for uniqueness of the solution. In the noisy case, when …
We address the inverse problem of Lagrangian identification based on trajecto-ries in the context of nonlinear optimal control. We propose a general formulation of the inverse problem based on occupation measures and complementarity in linear programming. The use of occupation measures in this context offers several advan-tages from the theoretical, numerical and statistical points of view. We propose an approximation procedure for which strong theoretical guarantees are available. Finally, the relevance of the method is illustrated on academic examples.
We study an inverse problem of the stochastic optimal control of general diffusions with performance index having the quadratic penalty term of the control process. Under mild conditions on the system dynamics, the cost functions, and the optimal control process, we show that our inverse problem is well-posed using a stochastic maximum principle. Then, with the well-posedness, we reduce the inverse problem to some root finding problem of the expectation of a random variable involved with the value function, which has a unique solution. Based on this result, we propose a numerical method for ou…
We propose a new clustering approach, called optimality-based clustering, that clusters data points based on their latent decision-making preferences. We assume that each data point is a decision generated by a decision-maker who (approximately) solves an optimization problem and cluster the data points by identifying a common objective function of the optimization problems for each cluster such that the worst-case optimality error is minimized. We propose three different clustering models and test them in the diet recommendation application.
The problem of attitude tracking using rotation matrices is addressed using an approach which combines inverse optimality and $\mathcal{L}_{2}$ disturbance attenuation. Conditions are provided which solve the inverse optimal nonlinear $H_{\infty}$ control problem by minimizing a meaningful cost function. The approach guarantees that the energy gain from an exogenous disturbance to a specified error signal respects a given upper bound. For numerical simulations, a simple problem setup from literature is considered and results demonstrate competitive performance.
We present a review of methods for optimal experimental design (OED) for Bayesian inverse problems governed by partial differential equations with infinite-dimensional parameters. The focus is on problems where one seeks to optimize the placement of measurement points, at which data are collected, such that the uncertainty in the estimated parameters is minimized. We present the mathematical foundations of OED in this context and survey the computational methods for the class of OED problems under study. We also outline some directions for future research in this area.
In this paper, the problem of finite horizon inverse optimal control (IOC) is investigated, where the quadratic cost function of a dynamic process is required to be recovered based on the observation of optimal control sequences. We propose the first complete result of the necessary and sufficient condition for the existence of corresponding LQ cost functions. Under feasible cases, the analytic expression of the whole solution space is derived and the equivalence of weighting matrices in LQ problems is discussed. For infeasible problems, an infinite dimensional convex problem is formulated to …
We describe an algorithm to solve Bellman optimization that replaces a sum over paths determining the optimal cost-to-go by an analytic method localized in state space. Our approach follows from the established relation between stochastic control problems in the class of linear Markov decision processes and quantum inverse scattering. We introduce a practical online computational method to solve for a potential function that informs optimal agent actions. This approach suggests that optimal control problems, including those with many degrees of freedom, can be solved with parallel computations…
We leverage second-order information for tuning of inverse optimal controllers for a class of discrete-time nonlinear input-affine systems. For this, we select the input penalty matrix, representing a tuning knob, to yield the Hessian of the Lyapunov function of the closed-loop dynamics. This draws a link between second-order methods known for their high speed of convergence and the tuning of inverse optimal stabilizing controllers to achieve a fast decay of the closed-loop trajectories towards a steady state. In particular, we ensure quadratic convergence, a feat that is otherwise not achieve…
Learning Universal Costs via Multi-Observation Inverse Optimal Transport
Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）；Department of Applied Mathematics, The Hong Kong Polytechnic University（香港理工大学应用数学系）
This paper revisits the well-studied \emph{optimal stopping} problem but within the \emph{large-population} framework. In particular, two classes of optimal stopping problems are formulated by taking into account the \emph{relative performance criteria}. It is remarkable the relative performance criteria, also understood by the \emph{Joneses preference}, \emph{habit formation utility}, or \emph{relative wealth concern} in economics and finance, plays an important role in explaining various decision behaviors such as price bubbles. By introducing such criteria in large-population setting, a giv…
In a recent paper, Nguyen, Kuhn, and Esfahani (2018) built a distributionally robust estimator for the precision matrix of the Gaussian distribution. The distributional uncertainty size is a key ingredient in the construction of this estimator. We develop a statistical theory which shows how to optimally choose the uncertainty size to minimize the associated Stein loss. Surprisingly, rather than the expected canonical square-root scaling rate, the optimal uncertainty size scales linearly with the sample size.
Inverse optimization describes a process that is the "reverse" of traditional mathematical optimization. Unlike traditional optimization, which seeks to compute optimal decisions given an objective and constraints, inverse optimization takes decisions as input and determines an objective and/or constraints that render these decisions approximately or exactly optimal. In recent years, there has been an explosion of interest in the mathematics and applications of inverse optimization. This paper provides a comprehensive review of both the methodological and application-oriented literature in inv…
In this paper, we define and solve the Inverse Stochastic Optimal Control (ISOC) problem of the linear-quadratic Gaussian (LQG) and the linear-quadratic sensorimotor (LQS) control model. These Stochastic Optimal Control (SOC) models are state-of-the-art approaches describing human movements. The LQG ISOC problem consists of finding the unknown weighting matrices of the quadratic cost function and the covariance matrices of the additive Gaussian noise processes based on ground truth trajectories observed from the human in practice. The LQS ISOC problem aims at additionally finding the covarianc…
We consider the problem of learning optimal solutions of a partially known linear optimization problem and recovering its underlying cost function where a set of past decisions and the feasible set are known. We develop a new framework, denoted as Inverse Learning, that extends the inverse optimization literature to (1) learn the optimal solution of the underlying problem, (2) integrate additional information on constraints and their importance, and (3) control the balance between mimicking past behaviors and reaching new goals and rules for the learned solution. We pose inverse learning as an…
We present a method to solve a special class of parameter identification problems for an elliptic optimal control problem to global optimality. The bilevel problem is reformulated via the optimal-value function of the lower-level problem. The reformulated problem is nonconvex and standard regularity conditions like Robinson's CQ are violated. Via a relaxation of the constraints, the problem can be decomposed into a family of convex problems and this is the basis for a solution algorithm. The convergence properties are analyzed. It is shown that a penalty method can be employed to solve this fa…
This study addresses some algorithms for solving structured unconstrained convex optimiza- tion problems using first-order information where the underlying function includes high-dimensional data. The primary aim is to develop an implementable algorithmic framework for solving problems with multi- term composite objective functions involving linear mappings using the optimal subgradient algorithm, OSGA, proposed by Neumaier in [49]. To this end, we propose some prox-functions for which the cor- responding subproblem of OSGA is solved in a closed form. Considering various inverse problems arisi…
Inverse optimal control with polynomial optimization
CNRS（法国国家科学研究中心；洛朗·阿利安系统分析实验室）；LAAS（图卢兹大学；洛朗·阿利安系统分析实验室）；Université de Toulouse（布拉格捷克理工大学电气工程学院）；LAAS；Faculty of Electrical Engineering, Czech Technical University in Prague
In this paper, we consider the problem of computing parameters of an objective function for a discrete-time optimal control problem from state and control trajectories with active control constraints. We propose a novel method of inverse optimal control that has a computationally efficient online form in which pairs of states and controls from given state and control trajectories are processed sequentially without being stored or processed in batches. We establish conditions guaranteeing the uniqueness of the objective-function parameters computed by our proposed method from trajectories with …
We introduce a new class of inverse optimization problems in which an input solution is given together with $k$ linear weight functions, and the goal is to modify the weights by the same deviation vector $p$ so that the input solution becomes optimal with respect to each of them, while minimizing $\|p\|_1$. In particular, we concentrate on three problems with multiple weight functions: the inverse shortest $s$-$t$ path, the inverse bipartite perfect matching, and the inverse arborescence problems. Using LP duality, we give min-max characterizations for the $\ell_1$-norm of an optimal deviation…
In a single facility location problem, a set of points is given and the goal is finding the optimal location of new facility respect to given criteria such as minimizing time, cost and distances between the clients and facilities. On the other side, the inverse models try to modify the parameters of the problem with the minimum cost such that a given point becomes optimal. In this paper, we introduce a novel algorithm for the general case of the inverse single facility location problem with variable weights in the plane. The convergence and optimality conditions of the algorithm are presented.…
This work deals with the event-triggered finite-time control for high-order systems based on an implicit Lyapunov function (ILF). With the construction of an inverse optimal problem, a novel expression of ILF is obtained. By designing the event-triggering mechanism elaborately, it is guaranteed that the trivial solution of the closed-loop system is globally finite-time stable and there exists no Zeno phenomenon. Extensions to the scenario with a multi-agent system are studied where a finite-time tracking control drives all the agents to reach a consensus. The obtained theoretical results are s…
We consider the maximum likelihood estimation of sparse inverse covariance matrices. We demonstrate that current heuristic approaches primarily encourage robustness, instead of the desired sparsity. We give a novel approach that solves the cardinality constrained likelihood problem to certifiable optimality. The approach uses techniques from mixed-integer optimization and convex optimization, and provides a high-quality solution with a guarantee on its suboptimality, even if the algorithm is terminated early. Using a variety of synthetic and real datasets, we demonstrate that our approach can …
CIIRC CTU in Prague（布拉格捷克理工大学信息、机器人与控制研究所）；Sorbonne Université, LIP6 CNRS（索邦大学，法国国家科学研究中心LIP6实验室）；LAAS-CNRS, FEE CTU in Prague（法国国家科学研究中心系统分析与架构实验室，布拉格捷克理工大学电气工程学院）
The Inverse Kinematics (IK) problem is to nd robot control parameters to bring it into the desired position under the kinematics and collision constraints. We present a global solution to the optimal IK problem for a general serial 7DOF manipulator with revolute joints and a quadratic polynomial objective function. We show that the kinematic constraints due to rotations can all be generated by second-degree polynomials. This is important since it signicantly simplies further step where we nd the optimal solution by Lasserre relaxations of non-convex polynomial systems. We demonstrate that the …
