Production scheduling with a piecewise-linear energy cost function

2016 ◽  
Author(s):  
Mustapha Haouassi ◽  
Chloe Desdouits ◽  
Rodolphe Giroudeau ◽  
Claude Le Pape
Keyword(s):  
Cost Function ◽  
Production Scheduling ◽  
Energy Cost ◽  
Piecewise Linear

Energy Cost Optimization of HVAC Loads Under Time-Varying Electricity Price Signals

2016 ◽  
Author(s):  
Saeid Bashash
Keyword(s):  
Dynamic Programming ◽  
Cost Function ◽  
Energy Cost ◽  
Optimal Operation ◽  
Electricity Price ◽  
Programming Approach ◽  
Time Varying ◽  
Temperature Deviation ◽  
Price Signals ◽  
The Cost

This paper presents a dynamic programming approach to optimize energy cost of multiple interacting household appliances such as air conditioning systems and refrigerators with temperature flexibility, under time varying electricity price signals. We adopt a first order differential equation model with a binary (ON-OFF) switching control function for each load. An energy cost minimization problem is then formulated with a pair of constraints on the temperature lower and upper bounds, as well as an equality condition on the initial and final temperature states. We use dynamic programming to compute cost-optimal control inputs and temperature trajectories for a given electricity price profile and ambient temperature condition. To account for temperature deviation from its desired setpoint, a quadratic temperature deviation penalty is added to the cost function. Moreover, to minimize the control input chattering for equipment protection, the cost function is expanded to also minimize the number of on-off switching events. Results for the different weighting combinations of the optimization objectives provide useful insights on the optimal operation of individual and multiple interacting HVAC loads. In particular, we observe that the loads are desynchronized under the cost-optimal operation, in the presence of local (renewable) power generation. The presented optimization algorithm and observed results can lead to the development of novel model predictive and rule-based feedback control policies for optimal energy management in households.

scholarly journals Proliferation of non-linear excitations in the piecewise-linear perceptron

2021 ◽  
Vol 10 (1) ◽  
Author(s):  
Antonio Sclocchi ◽  
Pierfrancesco Urbani
Keyword(s):  
Cost Function ◽  
Optimization Problem ◽  
Hard Spheres ◽  
Piecewise Linear ◽  
Power Laws ◽  
Universality Class ◽  
Local Minima ◽  
Convex Optimization Problem ◽  
Non Linear ◽  
The Cost

We investigate the properties of local minima of the energy landscape of a continuous non-convex optimization problem, the spherical perceptron with piecewise linear cost function and show that they are critical, marginally stable and displaying a set of pseudogaps, singularities and non-linear excitations whose properties appear to be in the same universality class of jammed packings of hard spheres. The piecewise linear perceptron problem appears as an evolution of the purely linear perceptron optimization problem that has been recently investigated in [1]. Its cost function contains two non-analytic points where the derivative has a jump. Correspondingly, in the non-convex/glassy phase, these two points give rise to four pseudogaps in the force distribution and this induces four power laws in the gap distribution as well. In addition one can define an extended notion of isostaticity and show that local minima appear again to be isostatic in this phase. We believe that our results generalize naturally to more complex cases with a proliferation of non-linear excitations as the number of non-analytic points in the cost function is increased.

scholarly journals A New Separable Piecewise Linear Learning Algorithm for the Stochastic Empty Container Repositioning Problem

2020 ◽  
Vol 2020 ◽  
pp. 1-16
Author(s):  
Shaorui Zhou ◽  
Xiaopo Zhuo ◽  
Zhiming Chen ◽  
Yi Tao
Keyword(s):  
Cost Function ◽  
Programming Model ◽  
Learning Algorithm ◽  
Piecewise Linear ◽  
Two Stage ◽  
Empty Container ◽  
Expected Cost ◽  
Stochastic Programs ◽  
Empty Container Repositioning ◽  
Empty Containers

A common challenge faced by liner operators in practice is to effectively allocate empty containers now in a way that minimizes the expectation of costs and reduces inefficiencies in the future with uncertainty. To incorporate uncertainties in the operational model, we formulate a two-stage stochastic programming model for the stochastic empty container repositioning (ECR) problem. This paper proposes a separable piecewise linear learning algorithm (SPELL) to approximate the expected cost function. The core of SPELL involves learning steps that provide information for updating the expected cost function adaptively through a sequence of piecewise linear separable approximations. Moreover, SPELL can utilize the network structure of the ECR problem and does not require any information about the distribution of the uncertain parameters. For the two-stage stochastic programs, we prove the convergence of SPELL. Computational results show that SPELL performs well in terms of operating costs. When the scale of the problem is very large and the dimensionality of the problem is increased, SPELL continues to provide consistent performance very efficiently and exhibits excellent convergence performance.

Optimizing Piecewise Linear Isolator for Steady State Vibration

2003 ◽  
Author(s):  
A. Narimani ◽  
M. F. Golnaraghi ◽  
R. N. Jazar
Keyword(s):  
Linear System ◽  
Cost Function ◽  
Frequency Response ◽  
Averaging Method ◽  
Piecewise Linear ◽  
Relative Displacement ◽  
Experimental Frequency ◽  
Piecewise Linear System ◽  
Transmitted Force ◽  
The Cost

In this work we have studied the effect of stoppers in quarter car model. While in rough roads these stoppers prevent the system from excessive displacement around the resonance frequency. Although stoppers prevent the undesired motion they increase the transmitted force that is undesirable in suspension systems. In order to optimize between the relative displacement and transmitted force an analytical model is considered. The piecewise linear system is the model of presented nonlinearity which cannot be considered small. Therefore standard perturbation methods are not able to provide an analytical solution. For this case we have adopted an averaging method which leads to frequency response of the piecewise linear system at resonance. In order to confirm the results obtained by averaging method an experimental device is fabricated and frequency response of the system is measured. The experimental frequency response is in very good agreement with analytical approach and numerical simulations. Sensitivity analysis methods are used to minimize the cost function of maximum relative displacement in frequency domain. The range of the parameters that minimizes the cost function where certain amount of clearance exists in the system are obtained.

scholarly journals Reconstruction of the Time-Dependent Volatility Function Using the Black–Scholes Model

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
Yuzi Jin ◽  
Jian Wang ◽  
Sangkwon Kim ◽  
Youngjin Heo ◽  
Changwoo Yoo ◽  
...  
Keyword(s):  
Cost Function ◽  
Piecewise Linear ◽  
Reconstruction Algorithm ◽  
Descent Method ◽  
Time Dependent ◽  
Steepest Descent Method ◽  
Market Values ◽  
Black Scholes ◽  
Volatility Function ◽  
The Cost

We propose a simple and robust numerical algorithm to estimate a time-dependent volatility function from a set of market observations, using the Black–Scholes (BS) model. We employ a fully implicit finite difference method to solve the BS equation numerically. To define the time-dependent volatility function, we define a cost function that is the sum of the squared errors between the market values and the theoretical values obtained by the BS model using the time-dependent volatility function. To minimize the cost function, we employ the steepest descent method. However, in general, volatility functions for minimizing the cost function are nonunique. To resolve this problem, we propose a predictor-corrector technique. As the first step, we construct the volatility function as a constant. Then, in the next step, our algorithm follows the prediction step and correction step at half-backward time level. The constructed volatility function is continuous and piecewise linear with respect to the time variable. We demonstrate the ability of the proposed algorithm to reconstruct time-dependent volatility functions using manufactured volatility functions. We also present some numerical results for real market data using the proposed volatility function reconstruction algorithm.

scholarly journals Optimal Control for a Class of Chaotic Systems

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Jianxiong Zhang ◽  
Wansheng Tang
Keyword(s):  
Optimal Control ◽  
Cost Function ◽  
Linear Systems ◽  
Chaotic Systems ◽  
Control Method ◽  
Piecewise Linear ◽  
Robust Optimal Control ◽  
Piecewise Linear Systems ◽  
Optimal Control Method ◽  
Optimal Control Methods

This paper proposes the optimal control methods for a class of chaotic systems via state feedback. By converting the chaotic systems to the form of uncertain piecewise linear systems, we can obtain the optimal controller minimizing the upper bound on cost function by virtue of the robust optimal control method of piecewise linear systems, which is cast as an optimization problem under constraints of bilinear matrix inequalities (BMIs). In addition, the lower bound on cost function can be achieved by solving a semidefinite programming (SDP). Finally, numerical examples are given to illustrate the results.

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