Degenerate Diffusion Processes

We present the experimental demonstration of the occurrence of exceptional points of degeneracy (EPDs) in a single resonator by introducing a linear time-periodic variation of one of its components. This is in contrast with the requirement of two coupled resonators with parity time-symmetric systems with precise values of gain and loss. In the proposed scheme, only the tuning of the modulation frequency is required, which is easily achieved in electronic systems. The EPD is a point in a system parameters’ space at which two or more eigenstates coalesce, and this leads to unique properties not occurring at other non-degenerate operating points. We show theoretically and experimentally the existence of a second-order EPD in a time-varying single resonator. Furthermore, we measure the sensitivity of the proposed system to a small structural perturbation and show that the two shifted system’s eigenfrequencies are well detected even for relative perturbations of 0.3%, with distinguished peaks well above the noise floor. We show that the regime of operation of the system at an EPD leads to a unique square-root-like sensitivity, which can devise new exceptionally sensitive sensors based on a single resonator by simply applying time modulation.

We consider a optimal dividend and stopping problem for two-dimensional compound Poisson risk model. The two companies allow to help each other by transffering payments. The goal of the companies is to maximize the expected cumulative discounted dividends payments up to the time of ruin. The problem is formulated as a singular control and optimal stopping problem. We explicitly construct the value function and the optimal strategy when claims are exponentially distributed and present two numerical examples.

This work generalizes the connection between the optimal singular stochastic control problem and the optimal stopping problem for regime-switching processes. Via the optimal singular stochastic control, the optimal stopping time and the continuation region are characterized. Moreover, we prove the existence of optimal singular stochastic control for a finite horizon singular control problem with the cost function containing the terminal cost. We prove it directly by the compactification method, which is based on an elaborate application of the properties of probability measures over the càdlàg space. Such a problem was left open in Haussmann and Suo [SIAM J. Control Optim., 33 (1995), pp. 916–936] and [SIAM J. Control Optim., 33 (1995), pp. 937–959]). In addition, our compactification method can remove the convexity condition on the coefficients used in Dufour and Miller [SIAM J. Control Optim., 43 (2004), pp. 708–730].

A singular stochastic control problem with state constraints in two-dimensions is studied. We show that the value function is $C^1$ and its directional derivatives are the value functions of certain optimal stopping problems. Guided by the optimal stopping problem, we then introduce the associated no-action region and the free boundary and show that, under appropriate conditions, an optimally controlled process is a Brownian motion in the no-action region with reflection at the free boundary. This proves a conjecture of Martins, Shreve and Soner [SIAM J. Control Optim. 34 (1996) 2133--2171] on the form of an optimal control for this class of singular control problems. An important issue in our analysis is that the running cost is Lipschitz but not $C^1$. This lack of smoothness is one of the key obstacles in establishing regularity of the free boundary and of the value function. We show that the free boundary is Lipschitz and that the value function is $C^2$ in the interior of the no-action region. We then use a verification argument applied to a suitable $C^2$ approximation of the value function to establish optimality of the conjectured control.

Motivated by theobservation—yet present in derivations of the Black-Scholes formula—that the value of a dynamic optimization problem is actually a stochastic process determined by the future development of the stochastic system, we reformulate a singular control problem as optimization problem for decoupled forward-backward SDE. The controlled process takes the form of a general Brownian diffusion in one dimension, with bounded variation control.Applications are given in the field of real investment and (real) options; many other optimization problems such as hedging with transaction costs or dividend control also exhibit a singular behaviour. Our reformulation enables us to consider nonadditive or stochastic differential utilities in the control problem, and exhibits formal similarities with reflected backward SDE and g-semisolutions.Related to the control problem we find a two-player stochastic game of optimal stopping (or Dynkin game). We establish a well-known relation between both problems for this general situation, namely that the partial derivative of the value of the control problem \( \frac{\partial } {{\partial x}}V \) equals the value of the Dynkin game u, and a saddle point for this game can be derived from an optimal control in the control problem. The arguments are based on stochastic analysis, especially comparison theorems for the solutions of controlled forward and backward stochastic differential equations.KeywordsControl ProblemStochastic Differential EquationComparison TheoremSingular ControlBackward Stochastic Differential EquationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Identifying comparative customer requirements from product online reviews for competitor analysis

We show that the value function of a singular stochastic control problem is equal to the integral of the value function of an associated optimal stopping problem. The connection is proved for a general class of diffusions using the method of viscosity solutions.

An active set algorithm (ASA) for box constrained optimization is developed. The algorithm consists of a nonmonotone gradient projection step, an unconstrained optimization step, and a set of rules for branching between the two steps. Global convergence to a stationary point is established. For a nondegenerate stationary point, the algorithm eventually reduces to unconstrained optimization without restarts. Similarly, for a degenerate stationary point, where the strong second-order sufficient optimality condition holds, the algorithm eventually reduces to unconstrained optimization without restarts. A specific implementation of the ASA is given which exploits the recently developed cyclic Barzilai-Borwein (CBB) algorithm for the gradient projection step and the recently developed conjugate gradient algorithm CG_DESCENT for unconstrained optimization. Numerical experiments are presented using box constrained problems in the CUTEr and MINPACK-2 test problem libraries.

Large scale optimization strategies have evolved considerably over the past two decades. Currently, continuous variable optimization problems (nonlinear programs) are solved on-line for steady state refinery models with several hundred thousand variables. Moreover, efficient NLP strategies have been developed for dynamic optimization problems. Still, to take the next step, on-line optimization of large dynamic chemical processes, a number of limitations and research challenges must be overcome. Many of the advances in NLP algorithms have taken place by recognizing and exploiting the framework of Successive Quadratic Programming (SQP) algorithms. These are extensions of Newton type methods for converging to the solution of the KKT (optimality) conditions of the optimization problem. Because of this, fast convergence can be expected and a number of standard devices can be added to stabilize the algorithm to converge from poor starting points. Limitations of these Newton-based methods are also well-known. They experience difficulties in the presence of ill conditioning and extreme nonlinearities. Also, for optimization algorithms, nonconvexity can also lead to a number of difficulties and there is a need for software that allows exploitable structures for specific problem classes. A number of innovations in algorithm design and problem formulation address these issues and greatly improve performance. As a result, very fast NLP algorithms can be derived for data reconciliation, parameter estimation, nonlinear model predictive control and dynamic optimization. Moreover, inequality constraints and variable bounds can be treated through advances in interior point strategies. These methods preserve the particular problem structure and scale well in performance for large-scale problems with many constraints. Finally, the ability to solve nonlinear programs quickly also allows us to consider more challenging problems. These include the extension to solving nonconvex NLPs globally and the ability to assess the solution’s tolerance to uncertainty by considering features of nonlinear programming sensitivity analysis and robust optimization through flexibility analysis and design.

Steel-making production is a dynamic process that has the characteristics of high temperature and heat, and a complex reaction mechanism that causes the mechanism model possibly to be unavailable and the system to be a black box. In this article, a dynamic operation optimization (DOO) problem is refined from the basic oxygen furnace (BOF) steel-making process, and the system model is formulated by a data analytics method. This prevents to solve the optimization problem with derivative-based optimization methods. To circumvent these difficulties, a surrogate-model-based derivative-free optimization algorithm is proposed for solving the DOO problem. In order to establish the surrogate model with the least number of function evaluations, a subset selection strategy is designed to find a sparse structure for the optimization problem, based on which a set of simple bases is determined to establish the surrogate model. Moreover, this also reduces the scale of the parameter optimization problem. Numerical experiments on actual production data verify the applicability and effectiveness of the proposed method.

In this paper, we study the behaviour of dynamic programming inequalities for optimal stopping and impulsive control of Markov processes when the generator of the uncontrolled process is - Aε such that $${A_\varepsilon } = A + {\varepsilon ^{ - 1}}B{\kern 1pt} and{\kern 1pt} \varepsilon \to o.$$

We establish the equivalence of competitive industry equilibrium with a central planner's decision problem under uncertainty, when investment is irreversible. The existence of industry equilibrium is derived, and it is shown that myopic behavior on the part of small agents is harmless, in the sense that it leads to the same decisions as full rational expectations do. Our model is set in continuous time and allows for very general forms of randomness. The methods are based on the probabilistic approach to singular stochastic control theory and its connections with optimal stopping problems.

Optimal stopping for Brownian motion with applications to sequential analysis and option pricing

We consider the problem of tracking a Brownian motion by a process of bounded variation, in such a way as to minimize total expected cost of both ‘action' and ‘deviation from a target state 0'. The former is proportional to the amount of control exerted to date, while the latter is being measured by a function which can be viewed, for simplicity, as quadratic. We discuss the discounted, stationary and finite-horizon variants of the problem. The answer to all three questions takes the form of exerting control in a singular manner, in order not to exit from a certain region. Explicit solutions are found for the first and second questions, while the third is reduced to an appropriate optimal stopping problem. This reduction yields properties, as well as global upper and lower bounds, for the associated moving boundary. The pertinent Abelian and ergodic relationships for the corresponding value functions are also derived.

Residential demand response is one of the key enabling technologies which plays an important role in managing the load demand of prosumers. However, the load scheduling problem becomes quite challenging due to the involvement of dynamic parameters and renewable energy resources. This work has proposed a bi-level load scheduling mechanism with dynamic electricity pricing integrated with renewable energy and storage system to overcome this problem. The first level involves the formulation of load scheduling and optimization problems as optimal stopping problems with the objective of energy consumption and delay cost minimization. This problem involved the real-time electricity pricing signal, customers load scheduling priority, machine learning (ML) based forecasted load demand, and renewable & storage unit profiles, which is solved using mathematical programming with branch-and-cut & branch-and-bound algorithms. Since the first-level optimization problem is formulated as a stopping problem, the optimal time slots are obtained using a one-step lookahead rule to schedule the load with the ability to handle the uncertainties. The second level is used to further model the load scheduling problem through the dynamic electricity pricing signal. The cost minimization objective function is then solved using the genetic algorithm (GA), where the input parameters are obtained from the first-level optimization solution. Furthermore, the impact of load prioritization in terms of time factor and electricity price is also modeled to allow the end-users to control their load. Analytical and simulation results are conducted using solar-home electricity data, Ausgrid, AUS to validate the proposed model. Results show that the proposed model can handle uncertainties involved in the load scheduling process along with a cost-effective solution in terms of cost and discomfort reduction. Furthermore, the bi-level process ensures cost minimization with end-user satisfaction regarding the dynamic electricity price signal.