This paper focuses on indefinite stochastic mean-field linear-quadratic (MF-LQ, for short) optimal control problems, which allow the weighting matrices for state and control in the cost functional to be indefinite. The solvability of stochastic Hamiltonian system and Riccati equations is presented under both positive definite case and indefinite case. The optimal controls in open-loop form and closed-loop form are obtained, respectively. Moreover, the dynamic mean-variance problem can be solved within the framework of the indefinite MF-LQ problem. Other two examples shed light on the theoretical results established.
This paper addresses an open problem in the area of linear quadratic optimal control. We consider the regular, infinite-horizon, stability-modulo-a-subspace, indefinite linear quadratic problem under the assumption that the dynamics are stabilizable. Our result generalizes previous works dealing with the same problem in the case of controllable dynamics. We explicitly characterize the unique solution of the algebraic Riccati equation that gives the optimal cost and optimal feedback control, as well as necessary and sufficient conditions for the existence of optimal controls.
In this paper, we consider a linear-quadratic optimal control problem of mean-field stochastic differential equation with jump diffusion, which is also called as an MF-LQJ problem. Here, cost functional is allowed to be indefinite. We use an equivalent cost functional method to deal with the MF-LQJ problem with indefinite weighting matrices. Some equivalent cost functionals enable us to establish a bridge between indefinite and positive-definite MF-LQJ problems. With such a bridge, solvabilities of stochastic Hamiltonian system and Riccati equations are further characterized. Optimal control o…
This paper is concerned with a backward stochastic linear-quadratic (LQ, for short) optimal control problem with deterministic coefficients. The weighting matrices are allowed to be indefinite, and cross-product terms in the control and state processes are present in the cost functional. Based on a Hilbert space method, necessary and sufficient conditions are derived for the solvability of the problem, and a general approach for constructing optimal controls is developed. The crucial step in this construction is to establish the solvability of a Riccati-type equation, which is accomplished und…
We present a generic approach that allows us to develop a fully polynomial-time approximation scheme (FTPAS) for minimizing nonlinear functions over the integer points in a rational polyhedron in fixed dimension. The approach combines the subdivision strategy of Papadimitriou and Yannakakis (2000) with ideas similar to those commonly used to derive real algebraic certificates of positivity for polynomials. Our general approach is widely applicable. We apply it, for instance, to the Motzkin polynomial and to indefinite quadratic forms $x^T Q x$ in a fixed number of variables, where $Q$ has at m…
A general backward stochastic linear-quadratic optimal control problem is studied, in which both the state equation and the cost functional contain the nonhomogeneous terms. The main feature of the problem is that the weighting matrices in the cost functional are allowed to be indefinite and cross-product terms in the control and the state processes are present. Necessary and sufficient conditions for the solvability of the problem are obtained, and a characterization of the optimal control in terms of forward-backward stochastic differential equations is derived. By a Riccati equation approac…
An abstract indefinite least squares problem with a quadratic constraint is considered. This is a quadratic programming problem with one quadratic equality constraint, where neither the objective nor the constraint are convex functions. Necessary and sufficient conditions are found for the existence of solutions.
An optimal ergodic control problem (EC problem, for short) is investigated for a linear stochastic differential equation with quadratic cost functional. Constant nonhomogeneous terms, not all zero, appear in the state equation, which lead to the asymptotic limit of the state non-zero. Under the stabilizability condition, for any (admissible) closed-loop strategy, an invariant measure is proved to exist, which makes the ergodic cost functional well-defined and the EC problem well-formulated. Sufficient conditions, including those allowing the weighting matrices of cost functional to be indefini…
It is proved that, for an indefinite quadratic programming problem under linear constraints, any iterative sequence generated by the Proximal DC decomposition algorithm $R$-linearly converges to a Karush-Kuhn-Tucker point, provided that the problem has a solution. Another major result of this paper says that DCA sequences generated by the algorithm converge to a locally unique solution of the problem if the initial points are taken from a suitably-chosen neighborhood of it. To deal with the implicitly defined iterative sequences, a local error bound for affine variational inequalities and nove…
Indefinite Stochastic Linear-Quadratic Optimal Control Problems with Random Coefficients and Poisson Jumps: Closed-Loop Representation of Open-Loop Optimal Controls
Southern University of Science and Technology（南方科技大学）；Southeast University（东南大学）
This paper studies uniform stabilization and social optimality for linear quadratic (LQ) mean field control problems with multiplicative noise, where agents are coupled via dynamics and individual costs. The state and control weights in cost functionals are not limited to be positive semi-definite. This leads to an indefinite LQ mean field control problem, which may still be well-posed due to deep nature of multiplicative noise. We first obtain a set of forward-backward stochastic differential equations (FBSDEs) from variational analysis, and construct a feedback control by decoupling the FBSD…
This paper is concerned with a stochastic linear-quadratic optimal control problem in a finite time horizon, where the coefficients of the control system are allowed to be random, and the weighting matrices in the cost functional are allowed to be random and indefinite. It is shown, with a Hilbert space approach, that for the existence of an open-loop optimal control, the convexity of the cost functional (with respect to the control) is necessary; and the uniform convexity, which is slightly stronger, turns out to be sufficient, which also leads to the unique solvability of the associated stoc…
Along this work we study an indefinite abstract smoothing problem. After establishing necessary and sufficient conditions for the existence of solutions to this problem, the set of admissible parameters is discussed in detail. Then, its relationship with a linearly constrained interpolation problem is analyzed.
A linear quadratic optimal stochastic control problem with random coefficients and indefinite state/control weight costs is usually linked to an indefinite stochastic Riccati equation (SRE) which is a matrix-valued quadratic backward stochastic differential equation along with an algebraic constraint involving the unknown. Either the optimal control problem or the SRE is solvable only if the given data satisfy a certain structure condition that has yet to be precisely defined. In this paper, by introducing a notion of subsolution for the SRE, we derive several novel sufficient conditions for t…
In standard linear quadratic (LQ) control, the first step in investigating infinite-horizon optimal control is to derive the stabilization condition with the optimal LQ controller. This paper focuses on the stabilization of an Ito stochastic system with indefinite control and state weighting matrices in the cost functional. A generalized algebraic Riccati equation (GARE) is obtained via the convergence of the generalized differential Riccati equation (GDRE) in the finite-horizon case. More importantly, the necessary and sufficient stabilization conditions for indefinite stochastic control are …
Algebraic Riccati equations with indefinite quadratic terms play an important role in applications related to robust controller design. While there are many established approaches to solve these in case of small-scale dense coefficients, there is no approach available to compute solutions in the large-scale sparse setting. In this paper, we develop an iterative method to compute low-rank approximations of stabilizing solutions of large-scale sparse continuous-time algebraic Riccati equations with indefinite quadratic terms. We test the developed approach for dense examples in comparison to oth…
In this paper, we give an algorithm that finds an epsilon-approximate solution to a mixed integer quadratic programming (MIQP) problem. The algorithm runs in polynomial time if the rank of the quadratic function and the number of integer variables are fixed. The running time of the algorithm is expected unless P=NP. In order to design this algorithm we introduce the novel concepts of spherical form MIQP and of aligned vectors, and we provide a number of results of independent interest. In particular, we give a strongly polynomial algorithm to find a symmetric decomposition of a matrix, and sho…
We introduce a coercivity condition as a time domain analogue of the frequency criterion provided by the famous Kalman-Yakubovich-Popov lemma. For a simple stochastic linear quadratic control problem we show how the coercivity condition characterizes the solvability of Riccati equations.
We consider policy gradient algorithms for the indefinite least squares stationary optimal control, e.g., linear-quadratic-regulator (LQR) with indefinite state and input penalization matrices. Such a setup has important applications in control design with conflicting objectives, such as linear quadratic dynamic games. We show the global convergence of gradient, natural gradient and quasi-Newton policies for this class of indefinite least squares problems.
This note concerns a class of matrix Riccati equations associated with stochastic linear-quadratic optimal control problems with indefinite state and control weighting costs. A novel sufficient condition of solvability of such equations is derived, based on a monotonicity property of a newly defined set. Such a set is used to describe a family of solvable equations.
This paper is concerned with the stochastic linear-quadratic optimal control problem with Poisson jumps. The coefficients in the state equation and the weighting matrices in the cost functional are all deterministic but are allowed indefinite. The notion of closed-loop strategies is introduced, and the optimal closed-loop strategy is characterized by a Riccati integral-differential equation and a backward stochastic differential equation with Poisson jumps.
This paper studies an asymptotic solvability problem for linear quadratic (LQ) mean field games with controlled diffusions and indefinite weights for the state and control in the costs. We employ a rescaling approach to derive a low dimensional Riccati ordinary differential equation (ODE) system, which characterizes a necessary and sufficient condition for asymptotic solvability. The rescaling technique is further used for performance estimates, establishing an $O(1/N)$-Nash equilibrium for the obtained decentralized strategies.
We study a special class of non-convex quadratic programs subject to two (possibly indefinite) quadratic constraints when the level sets of the constraint functions are {\it not} arranged {\it alternatively.} It is shown in the paper that this class of problems admit strong duality following a tight SDP relaxation, without assuming primal or dual Slater conditions. Our results cover Ye and Zhang's development in 2003 and the generalized trust region subproblems (GTRS) as special cases. Through the novel geometric view and some simple examples, we can explain why the problem becomes very hard w…
This paper considers linear-quadratic (LQ) stochastic leader-follower Stackelberg differential games for jump-diffusion systems with random coefficients. We first solve the LQ problem of the follower using the stochastic maximum principle and obtain the state-feedback representation of the open-loop optimal solution in terms of the integro-stochastic Riccati differential equation (ISRDE), where the state-feedback type control is shown to be optimal via the completion of squares method. Next, we establish the stochastic maximum principle for the indefinite LQ stochastic optimal control problem …
This paper revisits well-studied dynamic decisions of weakly coupled large-population (LP) systems. Specifically, three types of LP decision problems: mean-field game (MG), mean-field team (MT), and mean-field-type control (MC), are completely analyzed in a general stochastic linear-quadratic setting with controlled-diffusion in state dynamics and indefinite weight in cost functional. More importantly, interrelations among MG, MT and MC are systematically discussed; some relevant interesting findings are reported that may be applied to a structural analysis of general LP decisions.
Multidimensional indefinite stochastic Riccati equations and zero-sum stochastic linear-quadratic differential games with non-Markovian regime switching
The paper investigates the problem of fitting protein complexes into electron density maps. They are represented by high-resolution cryoEM density maps converted into overlapping matrices and partly show a structure of a complex. The general purpose is to define positions of all proteins inside it. This problem is known to be NP-hard, since it lays in the field of combinatorial optimization over a set of discrete states of the complex. We introduce quadratic programming approaches to the problem. To find an approximate solution, we convert a density map into an overlapping matrix, which is gen…
Inverse reinforcement learning for indefinite mean-field social optimization with multiplicative noise
School of Control Science and Engineering, Shandong University（山东大学控制科学与工程学院）；Department of Applied Mathematics, The Hong Kong Polytechnic University（香港理工大学应用数学系）
This paper is concerned with optimal control of stochastic fully coupled forward-backward linear quadratic (FBLQ) problems with indefinite control weight costs. In order to obtain the state feedback representation of the optimal control, we propose a new decoupling technique and obtain one kind of non-Riccati-type ordinary differential equations (ODEs). By applying the completion-of-squares method, we prove the existence of the solutions for the obtained ODEs under some assumptions and derive the state feedback form of the optimal control. For this FBLQ problem, the optimal control depends on …
It is well known that stability is the most fundamental nature with regard to a control system, in view of this, the stabilization becomes an inevitable control problem. This article mainly discusses the optimal control and stabilization problem for discrete-time systems involving Markov jump and multiplicative noise. The state and control weighting matrices in the cost function are allowed to be indefinite. By solving the forward-backward stochastic difference equations with Markov jump (FBSDEs-MJ) derived from the maximum principle, we conclude that the necessary and sufficient conditions of…
This paper studies asymptotic solvability of a linear quadratic (LQ) mean field social optimization problem with controlled diffusions and indefinite state and control weights. Starting with an $N$-agent model, we employ a rescaling approach to derive a low-dimensional Riccati ordinary differential equation (ODE) system, which characterizes a necessary and sufficient condition for asymptotic solvability. The decentralized control obtained from the mean field limit ensures a bounded optimality loss in minimizing the social cost having magnitude $O(N)$, which implies an optimality loss of $O(1/N…
Data-Driven Exploration for a Class of Continuous-Time Indefinite Linear--Quadratic Reinforcement Learning Problems
Department of Industrial Engineering and Operations Research, Columbia University（工业工程与运营管理系，哥伦比亚大学）；Data Science Institute, Columbia University（数据科学研究所，哥伦比亚大学）
针对连续时间随机 LQ 控制的无模型 RL，本文提出由 critic 与 actor 自适应调节熵正则化和策略方差的数据驱动探索机制，以少量调参实现匹配已知最优固定调度结果的次线性遗憾，并在实验中加速收敛、降低遗憾。
We study the closed-loop solvability of a stochastic linear quadratic optimal control problem for systems governed by stochastic evolution equations. This solvability is established by means of solvability of the corresponding Riccati equation, which is implied by the uniform convexity of the quadratic cost functional. At last, conditions ensuring the uniform convexity of the cost functional are discussed.
In this paper, we consider linear quadratic optimal control with mean-field type for discrete-time stochastic systems with state and control dependent noise. An optimal control problem is studied for a linear mean-field stochastic differential equation with a quadratic cost functional. The coefficients and the weighting matrices in the cost functional are all assumed to be deterministic.
This paper considers the stochastic linear quadratic optimal control problem in which the control domain is nonconvex. By the functional analysis and convex perturbation methods, we establish a novel maximum principle. The application of the proposed maximum principle is illustrated through a work-out example.
One of the fundamental issues in Control Theory is to design feedback controls. It is well-known that, the purpose of introducing Riccati equations in the deterministic case is to provide the desired feedback controls for linear quadratic control problems. To date, the same problem in the stochastic setting is only partially well-understood. In this paper, we establish the equivalence between the existence of optimal feedback controls for the stochastic linear quadratic control problems with random coefficients and the solvability of the corresponding backward stochastic Riccati equations in a…
