Optimal Multi Zones Search Technique to Detect a Lost Target by Using K Sensors

This paper presents the discrete search technique on multi zones to detect a lost target by using sensors. The search region is divided into zones. These zones contain an equal number of states (cells) not necessarily identical. Each zone has a one sensor to detect the target. The target moves over the cells according to a random process. We consider the searching effort as a random variable with a known probability distribution. The detection function with the discounted reward function in a certain state and time interval are given. The optimal effort distribution that minimizes the probability of undetection is obtained after solving a discrete stochastic optimization problem. An algorithm is constructed to obtain the optimal solution as in the numerical application.

Graphical Convergence of Subgradients in Nonconvex Optimization and Learning

We investigate the stochastic optimization problem of minimizing population risk, where the loss defining the risk is assumed to be weakly convex. Compositions of Lipschitz convex functions with smooth maps are the primary examples of such losses. We analyze the estimation quality of such nonsmooth and nonconvex problems by their sample average approximations. Our main results establish dimension-dependent rates on subgradient estimation in full generality and dimension-independent rates when the loss is a generalized linear model. As an application of the developed techniques, we analyze the nonsmooth landscape of a robust nonlinear regression problem.

Stochastic approximation method using diagonal positive-definite matrices for convex optimization with fixed point constraints

This paper proposes a stochastic approximation method for solving a convex stochastic optimization problem over the fixed point set of a quasinonexpansive mapping. The proposed method is based on the existing adaptive learning rate optimization algorithms that use certain diagonal positive-definite matrices for training deep neural networks. This paper includes convergence analyses and convergence rate analyses for the proposed method under specific assumptions. Results show that any accumulation point of the sequence generated by the method with diminishing step-sizes almost surely belongs to the solution set of a stochastic optimization problem in deep learning. Additionally, we apply the learning methods based on the existing and proposed methods to classifier ensemble problems and conduct a numerical performance comparison showing that the proposed learning methods achieve high accuracies faster than the existing learning method.

Value functions in a regime switching jump diffusion with delay market model

<abstract><p>In this paper, we consider a market model where the risky asset is a jump diffusion whose drift, volatility and jump coefficients are influenced by market regimes and history of the asset itself. Since the trajectory of the risky asset is discontinuous, we modify the delay variable so that it remains defined in this discontinuous setting. Instead of the actual path history of the risky asset, we consider the continuous approximation of its trajectory. With this modification, the delay variable, which is a sliding average of past values of the risky asset, no longer breaks down. We then use the resulting stochastic process in formulating the state variable of a portfolio optimization problem. In this formulation, we obtain the dynamic programming principle and Hamilton Jacobi Bellman equation. We also provide a verification theorem to guarantee the optimal solution of the corresponding stochastic optimization problem. We solve the resulting finite time horizon control problem and show that close form solutions of the stochastic optimization problem exist for the cases of power and logarithmic utility functions. In particular, we show that the HJB equation for the power utility function is a first order linear partial differential equation while that of the logarithmic utility function is a linear ordinary differential equation.</p></abstract>

On Optimal Leader’s Investments Strategy in a Cyclic Model of Innovation Race with Random Inventions Times

In this paper, we develop a new dynamic model of optimal investments in R&D and manufacturing for a technological leader competing with a large number of identical followers on the market of a technological product. The model is formulated in the form of the infinite time horizon stochastic optimization problem. The evolution of new generations of the product is treated as a Poisson-type cyclic stochastic process. The technology spillovers effect acts as a driving force of technological change. We show that the original probabilistic problem that the leader is faced with can be reduced to a deterministic one. This result makes it possible to perform analytical studies and numerical calculations. Numerical simulations and economic interpretations are presented as well.

Adaptive Submodular Ranking and Routing

Many applications of stochastic optimization involve making sequential decisions until some stopping criterion is satisfied. For example, in medical diagnosis, a doctor needs to perform an adaptive sequence of tests on a patient in order to diagnose a disease. Being adaptive allows the doctor to choose the next test based on the outcomes of prior tests. Given an a priori probability distribution over diseases, the goal is to minimize the expected cost of tests. In “Adaptive Submodular Ranking and Routing,” Navidi, Kambadur, and Nagarajan formulate a general stochastic optimization problem in which the stopping criterion corresponds to covering a submodular function. Such problems arise in many applications, including active learning, robotics, and disaster management. The authors obtain efficient algorithms with best possible performance guarantees. These results also extend to a vehicle-routing setting, in which one needs to plan an adaptive route based on information observed at nodes in the network. The authors also present experimental results on a data set arising in the identification of toxic chemicals, thereby demonstrating the practical applicability of their algorithm.

Hierarchical Model Predictive Control for Hydraulic Hybrid Powertrain of a Construction Vehicle

Hybrid hydraulic technology has the advantages of high-power density and low price and shows good adaptability in construction machinery. A complex hybrid powertrain architecture requires optimization and management of power demand distribution and an accurate response to desired power distribution of the power source subsystems in order to achieve target performances in terms of fuel consumption, drivability, component lifetime, and exhaust emissions. For hybrid hydraulic vehicles (HHVs) that are used in construction machinery, the challenge is to design an appropriate control scheme to actually achieve fuel economy improvement taking into consideration the relatively low energy density of the hydraulic accumulator and frequent load changes, the randomness of the driving conditions, and the uncertainty of the engine dynamics. To improve fuel economy and adaptability of various driving conditions to online energy management and to enhance the response performance of an engine to a desired torque, a hierarchical model predictive control (MPC) scheme is presented in this paper using the example of a spray-painting construction vehicle. The upper layer is a stochastic MPC (SMPC) based energy management control strategy (EMS) and the lower layer is an MPC-based tracking controller with disturbance estimator of the diesel engine. In the SMPC-EMS of the upper-layer management, a Markov model is built using driving condition data of the actual construction vehicle to predict future torque demands over a finite receding horizon to deal with the randomness of the driving conditions. A multistage stochastic optimization problem is formulated, and a scenario-based enumeration approach is used to solve the stochastic optimization problem for online implementation. In the lower-layer tracking controller, a disturbance estimator is designed to handle the uncertainty of the engine, and the MPC is introduced to ensure the tracking performance of the output torque of the engine for the distributed torque from the upper-layer SMPC-EMS, and therefore really achieve high efficiency of the diesel engine. The proposed strategy is evaluated using both simulation MATLAB/Simulink and the experimental test platform through a comparison with several existing strategies in two real driving conditions. The results demonstrate that the proposed strategy (SMPC+MPC) improves miles per gallon an average by 7.3% and 5.9% as compared with the control strategy (RB+PID) consisting of a rule-based (RB) management strategy and proportional-integral-derivative (PID) controller of the engine in simulation and experiment, respectively.

Optimal pricing structure in a maintenance contract: a game theory approach

This essay presents a novel look at Murthy and Asgharizadeh's study (Murthy & Asgharizadeh, 1998). The authors developed a decision problem applied to maintenance outsourcing involving two decision-makers (players). If a consumer buys a product, then outsources the maintenance actions to a maintenance agent (agent) who offers two maintenance options; a maintenance contract that holds a penalty clause which is activated if the agent's time to repair is higher than a specified time, and services on-demand. The model yields equilibrium strategies based on the subgame-perfect Nash equilibrium. The agent defines the optimal pricing structure for the maintenance options considering the equipment's useful life while the consumer maximizes their expected payoff by choosing one maintenance option. Our contribution to this research branches in three ways. First, once the model deals with random variables, it represents a stochastic optimization problem. We propose a different approach to estimate this penalty time by using the Monte Carlo method. The second contribution is to present a formal definition of this decision problem as a game, emphasizing the game theory's components. Finally, we reinterpret the players' equilibrium strategies.