This paper presents an inverse optimality method to solve the Hamilton-Jacobi-Bellman equation for a class of nonlinear problems for which the cost is quadratic and the dynamics are affine in the input. The method is inverse optimal because the running cost that renders the control input optimal is also explicitly determined. One special feature of this work, as compared to other methods in the literature, is the fact that the solution is obtained directly for the control input. The value function can also be obtained after one solves for the control input. Furthermore, a Lyapunov function that proves at least local stability of the controller is also obtained. In this regard the main contribution of this paper can be interpreted in two different ways: offering an analytical expression for Lyapunov functions for a class of nonlinear systems and obtaining an optimal controller for the same class of systems using a specific optimization functional. We also believe that an additional contribution of this paper is to identify explicit classes of systems and optimization functionals for which optimal control problems can be solved analytically. In particular, for second order systems three cases are identified: i) control input only as a function of the second state variable, ii) control input affine in the second state variable when the dynamics are affine in that variable and iii) control input affine in the first state variable when the dyamics are affine in that variable. The relevance of the proposed methodology is illustrated in several examples, including the Van der Pol oscillator, mass-spring systems and vehicle path following.
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.
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…
The selection problem of an optimal set of sensors estimating the snapshot of high-dimensional data is considered. The objective functions based on various criteria of optimal design are adopted to the greedy method: D-optimality, A-optimality, and E-optimality, which maximizes the determinant, minimize the trace of inverse, and maximize the minimum eigenvalue of the Fisher information matrix, respectively. First, the Fisher information matrix is derived depending on the numbers of latent state variables and sensors. Then, the unified formulation of the objective function based on A-optimality…
Disaggregated Benders decomposition and lazy constraints for solving the budget-constrained dynamic uncapacitated facility location and network design problem
Hidden convex optimization is such a class of nonconvex optimization problems that can be globally solved in polynomial time via equivalent convex programming reformulations. In this paper, we focus on checking local optimality in hidden convex optimization. We first introduce a class of hidden convex optimization problems by jointing the classical nonconvex trust-region subproblem (TRS) with convex optimization (CO), and then present a comprehensive study on local optimality conditions. In order to guarantee the existence of a necessary and sufficient condition for local optimality, we need m…
Inverse optimization, determining parameters of an optimization problem that render a given solution optimal, has received increasing attention in recent years. While significant inverse optimization literature exists for convex optimization problems, there have been few advances for discrete problems, despite the ubiquity of applications that fundamentally rely on discrete decision-making. In this paper, we present a new set of theoretical insights and algorithms for the general class of inverse mixed integer linear optimization problems. Specifically, a general characterization of optimality…
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.…
Optimality Conditions and Numerical Algorithms for a Class of Minimax Bilevel Optimization Problems
Xiangtan University（湘潭大学）；Beijing University of Posts and Telecommunications（北京邮电大学）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
Value iteration (VI) is a ubiquitous algorithm for optimal control, planning, and reinforcement learning schemes. Under the right assumptions, VI is a vital tool to generate inputs with desirable properties for the controlled system, like optimality and Lyapunov stability. As VI usually requires an infinite number of iterations to solve general nonlinear optimal control problems, a key question is when to terminate the algorithm to produce a "good" solution, with a measurable impact on optimality and stability guarantees. By carefully analysing VI under general stabilizability and detectabilit…
Converse optimality theory addresses an optimal control problem conversely where the system is unknown and the value function is chosen. Previous work treated this problem both in continuous and discrete time and non-extensively considered disturbances. In this paper, the converse optimality theory is extended to the class of affine systems with disturbances in continuous time while considering norm constraints on both control inputs and disturbances. The admissibility theorem and the design of the internal dynamics model are generalized in this context. A robust stabilizability condition is a…
We propose a supervised learning framework for computing solutions of multi-parametric Mixed Integer Linear Programs (MILPs) that arise in Model Predictive Control. Our approach also quantifies sub-optimality for the computed solutions. Inspired by Branch-and-Bound techniques, the key idea is to train a Neural Network/Random Forest, which for a given parameter, predicts a strategy consisting of (1) a set of Linear Programs (LPs) such that their feasible sets form a partition of the feasible set of the MILP and (2) a candidate integer solution. For control computation and sub-optimality quantif…
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.
In this paper, we propose a hybrid framework to solve large-scale permutation-based combinatorial problems effectively using a high-performance quadratic unconstrained binary optimization (QUBO) solver. To do so, transformations are required to change a constrained optimization model to an unconstrained model that involves parameter tuning. We propose techniques to overcome the challenges in using a QUBO solver that typically comes with limited numbers of bits. First, to smooth the energy landscape, we reduce the magnitudes of the input without compromising optimality. We propose a machine lea…
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…
Massachusetts Institute of Technology（麻省理工学院）；London Business School（伦敦商学院）
We propose a framework for modeling and solving low-rank optimization problems to certifiable optimality. We introduce symmetric projection matrices that satisfy $Y^2=Y$, the matrix analog of binary variables that satisfy $z^2=z$, to model rank constraints. By leveraging regularization and strong duality, we prove that this modeling paradigm yields tractable convex optimization problems over the non-convex set of orthogonal projection matrices. Furthermore, we design outer-approximation algorithms to solve low-rank problems to certifiable optimality, compute lower bounds via their semidefinite…
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
