Abstract:
a single product, multi-period, aggregate production planning problem is formulated as a linear-quadratic gaussian (lqg) optimal control problem with probabilistic constraints on state and control variables. this stochastic problem is based on a classical model developed by holt, modigliani, muth and simon, and known in the literature as hmms model. the central idea is to extend the original hmms model in order to take into account both chance-constraints on the decision variables and an arma forecasting model to represent the fluctuation of demand. essentially, the paper discusses the main features that allow transforming the problem into a chance constrained lqg pattern. in addition, two sub-optimal techniques for solving this kind of problem are briefly described. at last, an illustrative example of how to provide aggregate production plans from the proposed problem is presented.

Abstract:
In this two-part paper, we identify a broad class of decentralized output-feedback LQG systems for which the optimal control strategies have a simple intuitive estimation structure and can be computed efficiently. Roughly, we consider the class of systems for which the coupling of dynamics among subsystems and the inter-controller communication is characterized by the same directed graph. Furthermore, this graph is assumed to be a multitree, that is, its transitive reduction can have at most one directed path connecting each pair of nodes. In this first part, we derive sufficient statistics that may be used to aggregate each controller's growing available information. Each controller must estimate the states of the subsystems that it affects (its descendants) as well as the subsystems that it observes (its ancestors). The optimal control action for a controller is a linear function of the estimate it computes as well as the estimates computed by all of its ancestors. Moreover, these state estimates may be updated recursively, much like a Kalman filter.

Abstract:
Distributed control problems under some specific information constraints can be formulated as (possibly infinite dimensional) convex optimization problems. The underlying motivation of this work is to develop an understanding of the optimal decision making architecture for such problems. In this paper, we particularly focus on the N-player triangular LQG problems and show that the optimal output feedback controllers have attractive state space realizations. The optimal controller can be synthesized using a set of stabilizing solutions to 2N linearly coupled algebraic Riccati equations, which turn out to be easily solvable under reasonable assumptions.

Abstract:
We consider the Monge-Kantorovich transport problem in a purely measure theoretic setting, i.e. without imposing continuity assumptions on the cost function. It is known that transport plans which are concentrated on c-monotone sets are optimal, provided the cost function c is either lower semi-continuous and finite, or continuous and may possibly attain the value infty. We show that this is true in a more general setting, in particular for merely Borel measurable cost functions provided that {c=infty} is the union of a closed set and a negligible set. In a previous paper Schachermayer and Teichmann considered strongly c-monotone transport plans and proved that every strongly c-monotone transport plan is optimal. We establish that transport plans are strongly c-monotone if and only if they satisfy a "better" notion of optimality called robust optimality.

Abstract:
This paper is concerned with the Coherent Quantum Linear Quadratic Gaussian (CQLQG) control problem of finding a stabilizing measurement-free quantum controller for a quantum plant so as to minimize an infinite-horizon mean square performance index for the fully quantum closed-loop system. In comparison with the observation-actuation structure of classical controllers, the coherent quantum feedback is less invasive to the quantum dynamics and quantum information. Both the plant and the controller are open quantum systems whose dynamic variables satisfy the canonical commutation relations (CCRs) of a quantum harmonic oscillator and are governed by linear quantum stochastic differential equations (QSDEs). In order to correspond to such oscillators, these QSDEs must satisfy physical realizability (PR) conditions, which are organised as quadratic constraints on the controller matrices and reflect the preservation of CCRs in time. The CQLQG problem is a constrained optimization problem for the steady-state quantum covariance matrix of the plant-controller system satisfying an algebraic Lyapunov equation. We propose a gradient descent algorithm equipped with adaptive stepsize selection for the numerical solution of the problem. The algorithm finds a local minimum of the LQG cost over the parameters of the Hamiltonian and coupling operators of a stabilizing PR quantum controller, thus taking the PR constraints into account. A convergence analysis of the proposed algorithm is presented. A numerical example of a locally optimal CQLQG controller design is provided to demonstrate the algorithm performance.

Abstract:
The purpose of this note is to show that the solution to the Kantorovich optimal transportation problem is supported on a Lipschitz manifold, provided the cost is $C^{2}$ with non-singular mixed second derivative. We use this result to provide a simple proof that solutions to Monge's optimal transportation problem satisfy a change of variables equation almost everywhere.

Abstract:
Generating optimal plans in highly dynamic environments is challenging. Plans are predicated on an assumed initial state, but this state can change unexpectedly during plan generation, potentially invalidating the planning effort. In this paper we make three contributions: (1) We propose a novel algorithm for generating optimal plans in settings where frequent, unexpected events interfere with planning. It is able to quickly distinguish relevant from irrelevant state changes, and to update the existing planning search tree if necessary. (2) We argue for a new criterion for evaluating plan adaptation techniques: the relative running time compared to the "size" of changes. This is significant since during recovery more changes may occur that need to be recovered from subsequently, and in order for this process of repeated recovery to terminate, recovery time has to converge. (3) We show empirically that our approach can converge and find optimal plans in environments that would ordinarily defy planning due to their high dynamics.

Abstract:
Based on a recently developed notion of physical realizability for quantum linear stochastic systems, we formulate a quantum LQG optimal control problem for quantum linear stochastic systems where the controller itself may also be a quantum system and the plant output signal can be fully quantum. Such a control scheme is often referred to in the quantum control literature as "coherent feedback control.'' It distinguishes the present work from previous works on the quantum LQG problem where measurement is performed on the plant and the measurement signals are used as input to a fully classical controller with no quantum degrees of freedom. The difference in our formulation is the presence of additional non-linear and linear constraints on the coefficients of the sought after controller, rendering the problem as a type of constrained controller design problem. Due to the presence of these constraints our problem is inherently computationally hard and this also distinguishes it in an important way from the standard LQG problem. We propose a numerical procedure for solving this problem based on an alternating projections algorithm and, as initial demonstration of the feasibility of this approach, we provide fully quantum controller design examples in which numerical solutions to the problem were successfully obtained. For comparison, we also consider the case of classical linear controllers that use direct or indirect measurements, and show that there exists a fully quantum linear controller which offers an improvement in performance over the classical ones.

Abstract:
This paper slightly improves a classical result by Gangbo and McCann (1996) about the structure of optimal transport plans for costs that are concave functions of the Euclidean distance. Since the main difficulty for proving the existence of an optimal map comes from the possible singularity of the cost at $0$, everything is quite easy if the supports of the two measures are disjoint; Gangbo and McCann proved the result under the assumption $\mu(\spt(\nu))=0$; in this paper we replace this assumption with the fact that the two measures are singular to each other. In this case it is possible to prove the existence of an optimal transport map, provided the starting measure $\mu$ does not give mass to small sets (i.e. $(d\!-\!1)-$rectifiable sets). When the measures are not singular the optimal transport plan decomposes into two parts, one concentrated on the diagonal and the other being a transport map between mutually singular measures.

Abstract:
We present the first provably almost-optimal gossip-based algorithms for aggregate computation that are both time optimal and message-optimal. Given a $n$-node network, our algorithms guarantee that all the nodes can compute the common aggregates (such as Min, Max, Count, Sum, Average, Rank etc.) of their values in optimal $O(\log n)$ time and using $O(n \log \log n)$ messages. Our result improves on the algorithm of Kempe et al. \cite{kempe} that is time-optimal, but uses $O(n \log n)$ messages as well as on the algorithm of Kashyap et al. \cite{efficient-gossip} that uses $O(n \log \log n)$ messages, but is not time-optimal (takes $O(\log n \log \log n)$ time). Furthermore, we show that our algorithms can be used to improve gossip-based aggregate computation in sparse communication networks, such as in peer-to-peer networks. The main technical ingredient of our algorithm is a technique called {\em distributed random ranking (DRR)} that can be useful in other applications as well. DRR gives an efficient distributed procedure to partition the network into a forest of (disjoint) trees of small size. Our algorithms are non-address oblivious. In contrast, we show a lower bound of $\Omega(n\log n)$ on the message complexity of any address-oblivious algorithm for computing aggregates. This shows that non-address oblivious algorithms are needed to obtain significantly better message complexity. Our lower bound holds regardless of the number of rounds taken or the size of the messages used. Our lower bound is the first non-trivial lower bound for gossip-based aggregate computation and also gives the first formal proof that computing aggregates is strictly harder than rumor spreading in the address-oblivious model.