This paper develops a unified methodology for probabilistic analysis and optimal control design for jump diffusion processes defined by polynomials. For such systems, the evolution of the moments of the state can be described via a system of linear ordinary differential equations. Typically, however, the moments are not closed and an infinite system of equations is required to compute statistical moments exactly. Existing methods for stochastic analysis, known as closure methods, focus on approximating this infinite system of equations with a finite dimensional system. This work develops an alternative approach in which the higher order terms, which are approximated in closure methods, are viewed as inputs to a finite-dimensional linear control system. Under this interpretation, upper and lower bounds of statistical moments can be computed via convex linear optimal control problems with semidefinite constraints. For analysis of steady-state distributions, this optimal control problem reduces to a static semidefinite program. These same optimization problems extend automatically to stochastic optimal control problems. For minimization problems, the methodology leads to guaranteed lower bounds on the true optimal value. Furthermore, we show how an approximate optimal control strategy can be constructed from the solution of the semidefinite program. The results are illustrated using numerous examples.
Most methods for stochastic optimal control focus on dynamic programming @cite_28 @cite_12 or approximate dynamic programming @cite_15 . More closely related to this work, however, are relaxation methods. Unlike approximate dynamic programming, these methods provide bounds on the achievable optimal values. The most closely related methods focus on occupation measures @cite_9 @cite_32 @cite_16 . These works rely on the general insight that a convex relaxation of an optimization problem can be obtained by examining randomized strategies @cite_20 . Such relaxation methods apply to stochastic optimal control problems, but it is not clear that they can be used for analysis problems. In contrast, our relaxation methodology applies to both cases. A more removed relaxation method is studied in @cite_17 @cite_13 . These works use duality to relax the constraint that controllers depend causally on state information. Also related is the work of @cite_35 @cite_36 which uses a combination of moment dynamics and deterministic optimal control to find approximate solutions to stochastic optimal control problems. This work, however, only considers systems with closed moments. Furthermore, this work assumes a fixed parametric form for the controller, and does not obtain a relaxation of the original stochastic control problem.
@cite_13 @cite_35 @cite_28 @cite_36 @cite_9 @cite_32 @cite_15 @cite_16 @cite_20 @cite_12 @cite_17
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This paper considers stochastic optimization problems whose objective functions involve powers of random variables. For example, consider the classic Stochastic lp Load Balancing Problem (SLBp): There are @math machines and @math jobs, and known independent random variables @math decribe the load incurred on machine @math if we assign job @math to it. The goal is to assign each jobs to machines in order to minimize the expected @math -norm of the total load on the machines. While convex relaxations represent one of the most powerful algorithmic tools, in problems such as SLBp the main difficulty is to capture the objective function in a way that only depends on each random variable separately. We show how to capture @math -power-type objectives in such separable way by using the @math -function method, introduced by Lata a to relate the moment of sums of random variables to the individual marginals. We show how this quickly leads to a constant-factor approximation for very general subset selection problem with @math -moment objective. Moreover, we give a constant-factor approximation for SLBp, improving on the recent @math -approximation of [, SODA 18]. Here the application of the method is much more involved. In particular, we need to sharply connect the expected @math -norm of a random vector with the @math -moments of its marginals (machine loads), taking into account simultaneously the different scales of the loads that are incurred by an unknown assignment. This paper proposes a novel two-stage optimization method for robust model predictive control (RMPC) with Gaussian disturbance and state estimation error. Since the disturbance is unbounded, it is impossible to achieve zero probability of violating constraints. Our goal is to optimize the expected value of an objective function while limiting the probability of violating any constraints over the planning horizon (joint chance constraint). Prior arts include ellipsoidal relaxation approach [1] and particle control [2], but the former yields very conservative result and the latter is computationally intensive. Our approach divide the optimization problem into two stages; the upper-stage that optimizes risk allocation, and the lower-stage that optimizes control sequence with tightened constraints. The lower-stage is a regular convex optimization, such as linear programming or quadratic programming. The upper-stage is also convex, but the objective function is not always differentiable. We developed a fast descent algorithm for the upper-stage called iterative risk allocation (IRA), which yield much smaller suboptimality than ellipsoidal relaxation method while achieving a substantial speedup compared to and particle control. We develop a mathematical programming approach for the classicalPSPACE-hard restless bandit problem in stochastic optimization. We introduce a hierarchy ofN (whereN is the number of bandits) increasingly stronger linear programming relaxations, the last of which is exact and corresponds to the (exponential size) formulation of the problem as a Markov decision chain, while the other relaxations provide bounds and are efficiently computed. We also propose a priority-index heuristic scheduling policy from the solution to the firstorder relaxation, where the indices are defined in terms of optimal dual variables. In this way we propose a policy and a suboptimality guarantee. We report results of computational experiments that suggest that the proposed heuristic policy is nearly optimal. Moreover, the second-order relaxation is found to provide strong bounds on the optimal value. Robotic systems need to be able to plan control actions that are robust to the inherent uncertainty in the real world. This uncertainty arises due to uncertain state estimation, disturbances, and modeling errors, as well as stochastic mode transitions such as component failures. Chance-constrained control takes into account uncertainty to ensure that the probability of failure, due to collision with obstacles, for example, is below a given threshold. In this paper, we present a novel method for chance-constrained predictive stochastic control of dynamic systems. The method approximates the distribution of the system state using a finite number of particles. By expressing these particles in terms of the control variables, we are able to approximate the original stochastic control problem as a deterministic one; furthermore, the approximation becomes exact as the number of particles tends to infinity. This method applies to arbitrary noise distributions, and for systems with linear or jump Markov linear dynamics, we show that the approximate problem can be solved using efficient mixed-integer linear-programming techniques. We also introduce an important weighting extension that enables the method to deal with low-probability mode transitions such as failures. We demonstrate in simulation that the new method is able to control an aircraft in turbulence and can control a ground vehicle while being robust to brake failures. The problem of obtaining the maximum a posteriori estimate of a general discrete Markov random field (i.e., a Markov random field defined using a discrete set of labels) is known to be NP-hard. However, due to its central importance in many applications, several approximation algorithms have been proposed in the literature. In this paper, we present an analysis of three such algorithms based on convex relaxations: (i) LP-S: the linear programming (LP) relaxation proposed by Schlesinger (1976) for a special case and independently in (2001), (1998), and (2005) for the general case; (ii) QP-RL: the quadratic programming (QP) relaxation of Ravikumar and Lafferty (2006); and (iii) SOCP-MS: the second order cone programming (SOCP) relaxation first proposed by Muramatsu and Suzuki (2003) for two label problems and later extended by (2006) for a general label set. We show that the SOCP-MS and the QP-RL relaxations are equivalent. Furthermore, we prove that despite the flexibility in the form of the constraints objective function offered by QP and SOCP, the LP-S relaxation strictly dominates (i.e., provides a better approximation than) QP-RL and SOCP-MS. We generalize these results by defining a large class of SOCP (and equivalent QP) relaxations which is dominated by the LP-S relaxation. Based on these results we propose some novel SOCP relaxations which define constraints using random variables that form cycles or cliques in the graphical model representation of the random field. Using some examples we show that the new SOCP relaxations strictly dominate the previous approaches. When designing optimal controllers for any system, it is often the case that the true state of the system is unknown to the controller. Imperfect state information must be taken into account in the controller’s design in order to preserve its optimality. The same is true when performing reachability calculations. To estimate the probability that the state of a stochastic system reaches, or stays within, some set of interest in a given time horizon, it is necessary to find a controller that drives the system to that set with maximum probability, given the controller’s knowledge of the true state of the system. To date, little work has been done on stochastic reachability calculations with partially observable states. The work that has been done relies on converting the reachability optimization problem to one with an additive cost function, for which theoretical results are well known. Our approach is to preserve the multiplicative cost structure when deriving a sufficient statistic that reduces the problem to one of perfect state information. Our transformation includes a change of measure that simplifies the distribution of the sufficient statistic conditioned on its previous value. We develop a dynamic programming recursion for the solution of the equivalent perfect information problem, proving that the recursion is valid, an optimal solution exists, and results in the same solution as to the original problem. We also show that our results are equivalent to those for the reformulated additive cost problem, and so such a reformulation is not required. Many Stochastic Optimal Control (SOC) approaches rely on samples to either obtain an estimate of the value function or a linearisation of the underlying system model. However, these approaches typically neglect the fact that the accuracy of the policy update depends on the closeness of the resulting trajectory distribution to these samples. The greedy operator does not consider such closeness constraint to the samples. Hence, the greedy operator can lead to oscillations or even instabilities in the policy updates. Such undesired behaviour is likely to result in an inferior performance of the estimated policy. We reuse inspiration from the reinforcement learning community and relax the greedy operator used in SOC with an information theoretic bound that limits the ‘distance’ of two subsequent trajectory distributions in a policy update. The introduced bound ensures a smooth and stable policy update. Our method is also well suited for model-based reinforcement learning, where we estimate the system dynamics model from data. As this model is likely to be inaccurate, it might be dangerous to exploit the model greedily. Instead, our bound ensures that we generate new data in the vicinity of the current data, such that we can improve our estimate of the system dynamics model. We show that our approach outperforms several state of the art approaches on challenging simulated robot control tasks. In this work, we present a unified view on Markov random fields (MRFs) and recently proposed continuous tight convex relaxations for multilabel assignment in the image plane. These relaxations are far less biased toward the grid geometry than Markov random fields on grids. It turns out that the continuous methods are nonlinear extensions of the well-established local polytope MRF relaxation. In view of this result, a better understanding of these tight convex relaxations in the discrete setting is obtained. Further, a wider range of optimization methods is now applicable to find a minimizer of the tight formulation. We propose two methods to improve the efficiency of minimization. One uses a weaker, but more efficient continuously inspired approach as initialization and gradually refines the energy where it is necessary. The other one reformulates the dual energy enabling smooth approximations to be used for efficient optimization. We demonstrate the utility of our proposed minimization schemes in numerical experiments. Finally, we generalize the underlying energy formulation from isotropic metric smoothness costs to arbitrary nonmetric and orientation dependent smoothness terms. We cast the partially observable control problem as a fully observable underactuated stochastic control problem in belief space and apply standard planning and control techniques. One of the difficulties of belief space planning is modeling the stochastic dynamics resulting from unknown future observations. The core of our proposal is to define deterministic beliefsystem dynamics based on an assumption that the maximum likelihood observation (calculated just prior to the observation) is always obtained. The stochastic effects of future observations are modeled as Gaussian noise. Given this model of the dynamics, two planning and control methods are applied. In the first, linear quadratic regulation (LQR) is applied to generate policies in the belief space. This approach is shown to be optimal for linearGaussian systems. In the second, a planner is used to find locally optimal plans in the belief space. We propose a replanning approach that is shown to converge to the belief space goal in a finite number of replanning steps. These approaches are characterized in the context of a simple nonlinear manipulation problem where a planar robot simultaneously locates and grasps an object. We consider the task of obtaining the maximum a posteriori estimate of discrete pairwise random fields with arbitrary unary potentials and semimetric pairwise potentials. For this problem, we propose an accurate hierarchical move making strategy where each move is computed efficiently by solving an st-MINCUT problem. Unlike previous move making approaches, e.g. the widely used a-expansion algorithm, our method obtains the guarantees of the standard linear programming (LP) relaxation for the important special case of metric labeling. Unlike the existing LP relaxation solvers, e.g. interior-point algorithms or tree-reweighted message passing, our method is significantly faster as it uses only the efficient st-MINCUT algorithm in its design. Using both synthetic and real data experiments, we show that our technique outperforms several commonly used algorithms. We consider the information relaxation approach for calculating performance bounds for stochastic dynamic programs DPs. This approach generates performance bounds by solving problems with relaxed nonanticipativity constraints and a penalty that punishes violations of these nonanticipativity constraints. In this paper, we study DPs that have a convex structure and consider gradient penalties that are based on first-order linear approximations of approximate value functions. When used with perfect information relaxations, these penalties lead to subproblems that are deterministic convex optimization problems. We show that these gradient penalties can, in theory, provide tight bounds for convex DPs and can be used to improve on bounds provided by other relaxations, such as Lagrangian relaxation bounds. Finally, we apply these results in two example applications: first, a network revenue management problem that describes an airline trying to manage seat capacity on its flights; and second, an inventory management problem with lead times and lost sales. These are challenging problems of significant practical interest. In both examples, we compute performance bounds using information relaxations with gradient penalties and find that some relatively easy-to-compute heuristic policies are nearly optimal. In the stochastic knapsack problem, we are given a knapsack of size B, and a set of jobs whose sizes and rewards are drawn from a known probability distribution. However, we know the actual size and reward only when the job completes. How should we schedule jobs to maximize the expected total reward? We know O(1)-approximations when we assume that (i) rewards and sizes are independent random variables, and (ii) we cannot prematurely cancel jobs. What can we say when either or both of these assumptions are changed? The stochastic knapsack problem is of interest in its own right, but techniques developed for it are applicable to other stochastic packing problems. Indeed, ideas for this problem have been useful for budgeted learning problems, where one is given several arms which evolve in a specified stochastic fashion with each pull, and the goal is to pull the arms a total of B times to maximize the reward obtained. Much recent work on this problem focus on the case when the evolution of the arms follows a martingale, i.e., when the expected reward from the future is the same as the reward at the current state. What can we say when the rewards do not form a martingale? In this paper, we give constant-factor approximation algorithms for the stochastic knapsack problem with correlations and or cancellations, and also for budgeted learning problems where the martingale condition is not satisfied. Indeed, we can show that previously proposed LP relaxations have large integrality gaps. We propose new time-indexed LP relaxations, and convert the fractional solutions into distributions over strategies, and then use the LP values and the time ordering information from these strategies to devise a randomized adaptive scheduling algorithm. We hope our LP formulation and decomposition methods may provide a new way to address other correlated bandit problems with more general contexts. The multi-armed bandit (MAB) problem features the classical tradeoff between exploration and exploitation. The input specifies several stochastic arms which evolve with each pull, and the goal is to maximize the expected reward after a fixed budget of pulls. The celebrated work of , surveyed in [8], presumes a condition on the arms called the martingale assumption. In [9], A. obtained an LP-based 1 48-approximation for the problem with the martingale assumption removed. We improve the algorithm to a 4 27-approximation, with simpler analysis. Our algorithm also generalizes to the case of MAB superprocesses with (stochastic) multi-period actions. This generalization captures the framework introduced by Guha and Munagala in [11], and yields new results for their budgeted learning problems. Also, we obtain a (1 2 -- e)-approximation for the variant of MAB where preemption (playing an arm, switching to another arm, then coming back to the first arm) is not allowed. This contains the stochastic knapsack problem of Dean, Goemans, and Vondrak in [6] with correlated rewards, where we are given a knapsack of fixed size, and a set of jobs each with a joint distribution for its size and reward. The actual size and reward of a job can only be discovered in real-time as it is being scheduled, and the objective is to maximize expected reward before the knapsack size is exhausted. Our (1 2 -- e)-approximation improves the 1 16 and 1 8 approximations in [9] for correlated stochastic knapsack with cancellation and no cancellation, respectively, providing to our knowledge the first tight algorithm for these problems that matches the integrality gap of 2. We sample probabilities from an exponential-sized dynamic programming solution, whose existence is guaranteed by an LP projection argument. We hope this technique can also be applied to other dynamic programming problems which can be projected down onto a small LP. This paper considers an important class of convex programming (CP) problems, namely, the stochastic composite optimization (SCO), whose objective function is given by the summation of general nonsmooth and smooth stochastic components. Since SCO covers non-smooth, smooth and stochastic CP as certain special cases, a valid lower bound on the rate of convergence for solving these problems is known from the classic complexity theory of convex programming. Note however that the optimization algorithms that can achieve this lower bound had never been developed. In this paper, we show that the simple mirror-descent stochastic approximation method exhibits the best-known rate of convergence for solving these problems. Our major contribution is to introduce the accelerated stochastic approximation (AC-SA) algorithm based on Nesterov’s optimal method for smooth CP (Nesterov in Doklady AN SSSR 269:543–547, 1983; Nesterov in Math Program 103:127–152, 2005), and show that the AC-SA algorithm can achieve the aforementioned lower bound on the rate of convergence for SCO. To the best of our knowledge, it is also the first universally optimal algorithm in the literature for solving non-smooth, smooth and stochastic CP problems. We illustrate the significant advantages of the AC-SA algorithm over existing methods in the context of solving a special but broad class of stochastic programming problems. This paper addresses the optimal control problem known as the Linear Quadratic Regulator in the case when the dynamics are unknown. We propose a multi-stage procedure, called Coarse-ID control, that estimates a model from a few experimental trials, estimates the error in that model with respect to the truth, and then designs a controller using both the model and uncertainty estimate. Our technique uses contemporary tools from random matrix theory to bound the error in the estimation procedure. We also employ a recently developed approach to control synthesis called System Level Synthesis that enables robust control design by solving a convex optimization problem. We provide end-to-end bounds on the relative error in control cost that are nearly optimal in the number of parameters and that highlight salient properties of the system to be controlled such as closed-loop sensitivity and optimal control magnitude. We show experimentally that the Coarse-ID approach enables efficient computation of a stabilizing controller in regimes where simple control schemes that do not take the model uncertainty into account fail to stabilize the true system. When controlling dynamic systems, such as mobile robots in uncertain environments, there is a trade off between risk and reward. For example, a race car can turn a corner faster by taking a more challenging path. This paper proposes a new approach to planning a control sequence with a guaranteed risk bound. Given a stochastic dynamic model, the problem is to find a control sequence that optimizes a performance metric, while satisfying chance constraints i.e. constraints on the upper bound of the probability of failure. We propose a two-stage optimization approach, with the upper stage optimizing the risk allocation and the lower stage calculating the optimal control sequence that maximizes reward. In general, the upper-stage is a non-convex optimization problem, which is hard to solve. We develop a new iterative algorithm for this stage that efficiently computes the risk allocation with a small penalty to optimality. The algorithm is implemented and tested on the autonomous underwater vehicle (AUV) depth planning problem, and demonstrates a substantial improvement in computation cost and suboptimality, compared to the prior arts. The optimal control problem where the state is governed by an Ito stochastic differential equation (possibly just an ordinary differential equation) is formulated in martingale terms. Under a coercivity condition (which is weaker than compactness of the control set), a convexity condition, and mild continuity hypotheses on the data, it is shown by the direct method that optimal controls exist. Hard and soft constraints are allowed. In the absence of soft constraints it is shown that there exists an optimal control that is a function only of the present time and state, i.e., the synthesis problem has a solution. The main tool here is Krylov’s Markovian Selection Theorem. Autonomous vehicles require optimal path planning algorithms to achieve mission goals while avoiding obstacles and being robust to uncertainties. The uncertainties arise from exogenous disturbances, modeling errors, and sensor noise, which can be characterized via stochastic models. Previous work dened a notion of robustness in a stochastic setting by using the concept of chance constraints. This requires that mission constraint violation can occur with a probability less than a prescribed value. In this paper we describe a novel method for optimal chance constrained path planning with feedback design. The approach optimizes both the reference trajectory to be followed and the feedback controller used to reject uncertainty. Our method extends recent results in constrained control synthesis based on convex optimization to solve control problems with nonconvex constraints. This extension is essential for path planning problems, which inherently have nonconvex obstacle avoidance constraints. Unlike previous approaches to chance constrained path planning, the new approach optimizes the feedback gain as well as the reference trajectory. The key idea is to couple a fast, nonconvex solver that does not take into account uncertainty, with existing robust approaches that apply only to convex feasible regions. By alternating between robust and nonrobust solutions, the new algorithm guarantees convergence to a global optimum. We apply the new method to an unmanned aircraft and show simulation results that demonstrate the ecacy of the approach. The usual approach to developing and analyzing first-order methods for non-smooth (stochastic or deterministic) convex optimization assumes that the objective function is uniformly Lipschitz continuous with parameter @math . However, in many settings the non-differentiable convex function @math is not uniformly Lipschitz continuous -- for example (i) the classical support vector machine (SVM) problem, (ii) the problem of minimizing the maximum of convex quadratic functions, and even (iii) the univariate setting with @math . Herein we develop a notion of "relative continuity" that is determined relative to a user-specified "reference function" @math (that should be computationally tractable for algorithms), and we show that many non-differentiable convex functions are relatively continuous with respect to a correspondingly fairly-simple reference function @math . We also similarly develop a notion of "relative stochastic continuity" for the stochastic setting. We analysis two standard algorithms -- the (deterministic) mirror descent algorithm and the stochastic mirror descent algorithm -- for solving optimization problems in these two new settings, and we develop for the first time computational guarantees for instances where the objective function is not uniformly Lipschitz continuous. This paper is a companion paper for non-differentiable convex optimization to the recent paper by Lu, Freund, and Nesterov, which developed similar sorts of results for differentiable convex optimization. We present an iterative linear-quadratic-Gaussian method for locally-optimal feedback control of nonlinear stochastic systems subject to control constraints. Previously, similar methods have been restricted to deterministic unconstrained problems with quadratic costs. The new method constructs an affine feedback control law, obtained by minimizing a novel quadratic approximation to the optimal cost-to-go function. Global convergence is guaranteed through a Levenberg-Marquardt method; convergence in the vicinity of a local minimum is quadratic. Performance is illustrated on a limited-torque inverted pendulum problem, as well as a complex biomechanical control problem involving a stochastic model of the human arm, with 10 state dimensions and 6 muscle actuators. A Matlab implementation of the new algorithm is availabe at www.cogsci.ucsd.edu spl sim todorov.