Quadratic Cost Research Articles

Accuracy study of linearization methods for quadratic cost curves of thermal units in unit commitment problems

To solve the unit commitment (UC) problems with efficient mixed-integer linear programming solvers, the quadratic cost functions (QCFs) of thermal units are always approximated to piecewise linear (PWL) functions. This study examines the accuracies of different approximation methods for piecewise linearizing the QCFs of units in UC problems. We use five piecewise linearization methods—evenly spaced PWL interpolation, evenly spaced PWL tangent, evenly spaced PWL–ɛmax/2 shifted interpolation, tighter PWL interpolation, and evenly spaced PWL least-squares fit—to approximate the QCFs of units. The authors first perform a series of reproductivity studies to verify the program. Then, numerical tests are conducted using different methods on the systems with 10, 100, and 800 units. The results show that different approximation methods lead to considerable differences in operating costs and the tighter PWL interpolation, compared with the other methods, is preferred in terms of approximation accuracy.

FoodMatch: Batching and Matching for Food Delivery in Dynamic Road Networks

Food delivery, today, is a multi-billion dollar industry. Minimizing food delivery time is a key contributor towards building positive customer experiences. More precisely, given a stream of food orders and available delivery vehicles, how should orders be assigned to vehicles so the delivery time is minimized? Several decisions have to be made: (1) assignment of orders to vehicles, (2) grouping orders into batches to cope with limited vehicle availability, (3) adapting to dynamic positions of delivery vehicles, and (4) ensuring scalability to the demands of real-world workloads. We show that the minimization problem is not only NP-hard but inapproximable in polynomial time. To mitigate this computational bottleneck, we develop an algorithm called FoodMatch , which maps the vehicle assignment problem to that of minimum weight perfect matching on a bipartite graph. To further reduce the quadratic construction cost of the bipartite graph, we deploy best-first search to only compute a subgraph that is highly likely to contain the minimum matching. The solution quality is further enhanced by reducing batching to a graph batching problem and anticipating dynamic positions of vehicles through angular distance . We perform extensive experiments on real food-delivery data from large metropolitan cities. Our results establish that FoodMatch imparts substantial improvements over baseline strategies across a host of metrics such as food delivery time, waiting time at restaurants, and orders delivered per kilometer. Furthermore, FoodMatch is efficient enough to handle real-world workloads.

Guaranteed Cost Impulsive Synchronization of Uncertain Multiplex Networks

This brief mainly deals with the issue of guaranteed cost impulsive synchronization of uncertain multi-agent systems from the multiplex network prospective, where the uncertain time-varying system parameters are supposed to be norm-bounded. By means of a group of linear matrix inequalities, the effective guaranteed cost impulsive control is designed to not only ensure that the trajectories of all followers synchronize with the leader, but also guarantee that the control cost is no higher than an upper bound of the quadratic guaranteed cost function. Numerical example demonstrates the feasibility of theoretical results.

BAYESIAN MITIGATION OF SPATIAL COARSENING FOR A HAWKES MODEL APPLIED TO GUNFIRE, WILDFIRE AND VIRAL CONTAGION.

Self-exciting spatiotemporal Hawkes processes have found increasing use in the study of large-scale public health threats, ranging from gun violence and earthquakes to wildfires and viral contagion. Whereas many such applications feature locational uncertainty, that is, the exact spatial positions of individual events are unknown, most Hawkes model analyses to date have ignored spatial coarsening present in the data. Three particular 21st century public health crises-urban gun violence, rural wildfires and global viral spread-present qualitatively and quantitatively varying uncertainty regimes that exhibit: (a) different collective magnitudes of spatial coarsening, (b) uniform and mixed magnitude coarsening, (c) differently shaped uncertainty regions and-less orthodox-(d) locational data distributed within the "wrong" effective space. We explicitly model such uncertainties in a Bayesian manner and jointly infer unknown locations together with all parameters of a reasonably flexible Hawkes model, obtaining results that are practically and statistically distinct from those obtained while ignoring spatial coarsening. This work also features two different secondary contributions: first, to facilitate Bayesian inference of locations and background rate parameters, we make a subtle yet crucial change to an established kernel-based rate model, and second, to facilitate the same Bayesian inference at scale, we develop a massively parallel implementation of the model's log-likelihood gradient with respect to locations and thus avoid its quadratic computational cost in the context of Hamiltonian Monte Carlo. Our examples involve thousands of observations and allow us to demonstrate practicality at moderate scales.

Network-Flow-Based Formulations for Convex Hull Pricing With Maximum Start-Ups

Reducing uplift payments has been a challenging problem for most wholesale markets in the U.S. The main difficulty comes from the discrete decision-making and non-linear objective function in the unit commitment (UC) problem. Recently, the convex hull pricing approach has shown promises to reduce the uplift payments, in which efficient algorithms are available when i) the convex hull description for each unit and ii) the convex envelope for the objective function are available. Following this framework, in this paper, we provide network-flow-based compact extended formulations for each UC considering ramping constraints, different initial statuses, and maximum start-up restrictions. Meanwhile, by using a piece-wise linear approximation of a quadratic cost curve in the objective function, we show that the corresponding piece-wise linear convex envelope converges to the quadratic envelope as the number of pieces increases in the convex hull pricing problem. The final computational experiments on a revised IEEE 118-bus system indicates the effectiveness of the proposed approach. The trade-off between the cost-saving and computational time improvement is also reported as a reference for further usage.

Signaling Games for Log-Concave Distributions: Number of Bins and Properties of Equilibria

We investigate the equilibrium behavior for the decentralized cheap talk problem for real random variables and quadratic cost criteria in which an encoder and a decoder have misaligned objective functions. In prior work, it has been shown that the number of bins in any equilibrium has to be countable, generalizing a classical result due to Crawford and Sobel who considered sources with density supported on [0, 1]. In this paper, we first refine this result in the context of log-concave sources. For sources with two-sided unbounded support, we prove that, for any finite number of bins, there exists a unique equilibrium. In contrast, for sources with semi-unbounded support, there may be a finite upper bound on the number of bins in equilibrium depending on certain conditions stated explicitly. Moreover, we prove that for log-concave sources, the expected costs of the encoder and the decoder in equilibrium decrease as the number of bins increases. Furthermore, for strictly log-concave sources with two-sided unbounded support, we prove convergence to the unique equilibrium under best response dynamics which starts with a given number of bins, making a connection with the classical theory of optimal quantization and convergence results of Lloyd’s method. In addition, we consider more general sources which satisfy certain assumptions on the tail(s) of the distribution and we show that there exist equilibria with infinitely many bins for sources with two-sided unbounded support. Further explicit characterizations are provided for sources with exponential, Gaussian, and compactly-supported probability distributions.

Fixed Beamline Optimization for Intensity Modulated Carbon-Ion Therapy.

A major obstacle for the adoption of heavy ion therapy is the cost and technical difficulties to construct and maintain a rotational gantry. Many heavy ion treatment facilities instead choose to construct fixed beamlines as a compromise, which we propose to mitigate with optimized treatment couch angle. We formulate the integrated beam orientation and scanning spot optimization problem as a quadratic cost function with a group sparsity regularization term. The optimization problem is efficiently solved using fast iterative shrinkage-thresholding algorithm (FISTA). To test the method, we created the fixed beamline plans with couch rotation (FBCR) and without couch rotation (FB) for intensity modulated carbon-ion therapy (IMCT) and compared with the ideal scenario where both the couch and gantry have 360 degrees of freedom (GCR). FB, FBCR, and GCR IMCT plans were compared for ten pancreas cases. The FBCR plans show comparable PTV coverage and OAR doses for each pancreas case. In conclusion, the dosimetric limitation of fixed beams in heavy ion radiotherapy may be largely mitigated with integrated beam orientation optimization of the couch rotation.

Bayesian Persuasion With State-Dependent Quadratic Cost Measures

In this article, we address Bayesian persuasion between a sender and a receiver with state-dependent quadratic cost measures for general classes of distributions. The receiver seeks to make mean-square-error estimate of a state based on a signal sent by the sender while the sender signals strategically in order to control the receiver’s estimate in a certain way. Such a scheme could model, e.g., deception and privacy, problems in multiagent systems. Existing solution concepts are not viable since here the receiver has continuous action space. We show that for finite state spaces, optimal signaling strategies can be computed through an equivalent linear optimization problem over the cone of completely positive matrices. We then establish its strong duality to a copositive program. To exemplify the effectiveness of this equivalence result, we adopt sequential polyhedral approximation of completely positive cones and analyze its performance numerically. We also quantify the approximation error for a quantized version of a continuous distribution and show that a semi-definite program relaxation of the equivalent problem could be a benchmark lower bound for the sender’s cost for large state spaces.

Mixed H∞ and passivity guaranteed cost proportional‐integral control for interval‐valued type‐2 fuzzy systems with input delay

AbstractThe paper focuses on the problem of mixed and passivity guaranteed cost control of an interval‐valued type‐2 (IT‐2) fuzzy system with time‐varying input delay and exogenous disturbance. For this, an IT‐2 fuzzy‐model‐based system with a mismatched membership grade function is formulated. The main objective is to design a proportional‐integral controller that stimulates our system to attain its equilibrium state even with the occupancy of exogenous disturbance and input delay in the system. In addition, a quadratic cost function is constructed such that the cost function of the closed‐loop IT‐2 fuzzy system is lower than the upper‐bound cost index. Mixed and passivity performance indices are then calculated to address the exogenous disturbance in the system. A set of new criteria ensuring the asymptotic stability of the closed‐loop IT‐2 fuzzy system with input delay are derived in terms of linear matrix inequalities with the aid of Lyapunov stability theory and the employment of a Wirtinger‐based inequality. Finally, two example simulations are conducted to validate our theoretical results.

Dynamic Regret Minimization for Control of Non-stationary Linear Dynamical Systems

We consider the problem of controlling a Linear Quadratic Regulator (LQR) system over a finite horizon T with fixed and known cost matrices Q,R, but unknown and non-stationary dynamics A_t, B_t. The sequence of dynamics matrices can be arbitrary, but with a total variation, V_T, assumed to be o(T) and unknown to the controller. Under the assumption that a sequence of stabilizing, but potentially sub-optimal controllers is available for all t, we present an algorithm that achieves the optimal dynamic regret of O(V_T^2/5 T^3/5 ). With piecewise constant dynamics, our algorithm achieves the optimal regret of O(sqrtST ) where S is the number of switches. The crux of our algorithm is an adaptive non-stationarity detection strategy, which builds on an approach recently developed for contextual Multi-armed Bandit problems. We also argue that non-adaptive forgetting (e.g., restarting or using sliding window learning with a static window size) may not be regret optimal for the LQR problem, even when the window size is optimally tuned with the knowledge of $V_T$. The main technical challenge in the analysis of our algorithm is to prove that the ordinary least squares (OLS) estimator has a small bias when the parameter to be estimated is non-stationary. Our analysis also highlights that the key motif driving the regret is that the LQR problem is in spirit a bandit problem with linear feedback and locally quadratic cost. This motif is more universal than the LQR problem itself, and therefore we believe our results should find wider application.

High-dimensional optimization under nonconvex excluded volume constraints.

We consider high-dimensional random optimization problems where the dynamical variables are subjected to nonconvex excluded volume constraints. We focus on the case in which the cost function is a simple quadratic cost and the excluded volume constraints are modeled by a perceptron constraint satisfaction problem. We show that depending on the density of constraints, one can have different situations. If the number of constraints is small, one typically has a phase where the ground state of the cost function is unique and sits on the boundary of the island of configurations allowed by the constraints. In this case, there is a hypostatic number of marginally satisfied constraints. If the number of constraints is increased one enters a glassy phase where the cost function has many local minima sitting again on the boundary of the regions of allowed configurations. At the phase transition point, the total number of marginally satisfied constraints becomes equal to the number of degrees of freedom in the problem and therefore we say that these minima are isostatic. We conjecture that by increasing further the constraints the system stays isostatic up to the point where the volume of available phase space shrinks to zero. We derive our results using the replica method and we also analyze a dynamical algorithm, the Karush-Kuhn-Tucker algorithm, through dynamical mean-field theory and we show how to recover the results of the replica approach in the replica symmetric phase.

Generalizations of Talagrand Inequality for Sinkhorn Distance Using Entropy Power Inequality.

The distance that compares the difference between two probability distributions plays a fundamental role in statistics and machine learning. Optimal transport (OT) theory provides a theoretical framework to study such distances. Recent advances in OT theory include a generalization of classical OT with an extra entropic constraint or regularization, called entropic OT. Despite its convenience in computation, entropic OT still lacks sufficient theoretical support. In this paper, we show that the quadratic cost in entropic OT can be upper-bounded using entropy power inequality (EPI)-type bounds. First, we prove an HWI-type inequality by making use of the infinitesimal displacement convexity of the OT map. Second, we derive two Talagrand-type inequalities using the saturation of EPI that corresponds to a numerical term in our expressions. These two new inequalities are shown to generalize two previous results obtained by Bolley et al. and Bai et al. Using the new Talagrand-type inequalities, we also show that the geometry observed by Sinkhorn distance is smoothed in the sense of measure concentration. Finally, we corroborate our results with various simulation studies.

Security tracking control for discrete-time stochastic systems subject to cyber attacks

This paper is concerned with the security tracking problem under the quadratic cost criterion for a class of discrete-time stochastic linear networked control systems (NCSs) exposed to cyber attacks, covering false data injection attacks as well as a class of DoS attacks. Taking into account factors such as network congestion and the defensive role of intrusion detection systems, successful attack events are modeled as a Bernoulli random sequence. To describe the transient trajectory of an NCS under the impact of a random attack, a probabilistic definition of secure trackability is taken. Therefore, an observer-based dynamic output feedback controller is designed in order to achieve the specified probabilistic secure trackability. Specifically, the probabilistic safety output tracking problem is transformed into an input-to-state stability problem in the probabilistic sense for closed-loop systems with some new sufficient conditions, provided that an augmented incremental model is utilized Then, the controller parameters and the upper bound of the quadratic cost function are determined by solving matrix inequalities, while easy-solution forms of the matrix inequalities to be solved are presented by the Schur complementary lemma. Both simulation studies and practical experiments demonstrate the effectiveness of the proposed control scheme.

Can sticky portfolios explain international capital flows and asset prices?

Recently portfolio choice has become an important element of many DSGE open economy models. Yet, a substantial body of evidence is inconsistent with standard frictionless portfolio choice models. In this paper we introduce a quadratic cost of changes in portfolio allocation into a two-country DSGE model. We investigate the level of portfolio frictions most consistent with the data and the impact of portfolio frictions on asset prices and net capital flows. We find the portfolio friction accounts for (i) micro evidence of portfolio inertia by households, (ii) macro evidence of the price impact of financial shocks and related disconnect of asset prices from fundamentals, (iii) a broad set of moments related to the time series behavior of saving, investment and net capital flows, and (iv) other phenomena relating to excess return dynamics. Financial and saving shocks each account for close to half of the variance of net capital flows.

Optimal reset law design based on guaranteed cost control method for Lipschitz nonlinear systems

AbstractThis article proposes a systematic approach for optimal reset law design of a class of nonlinear systems. By using the guaranteed cost control method, sufficient conditions for the design of optimal reset law are derived in terms of linear matrix inequalities. In an offline design procedure, the reset law is computed that minimizes the upper bound of a quadratic cost function. The proposed method can be implemented for real‐time applications even with small sampling time. The simulation results verify the efficacy and effectiveness of the proposed theoretical results.

Stochastic event-based LQG control: An analysis on strict consistency

This paper studies the discrete-time linear–quadratic-Gaussian problem under partial-information-based sensors and event-based transmissions. By constructing a special form of stochastic event-based protocols, the optimal control policy is obtained, which minimizes an average quadratic cost function under a fixed transmission protocol. Then, analytical expressions of the corresponding transmission and control performance are derived. To evaluate the efficiency of event-based protocols, the concept of strict consistency is introduced, which requires that the control performance of event-based protocols be strictly better than that of the periodic protocol under the same transmission rate. Based on the obtained analytical expressions, the strict consistency is proved by showing that the periodic protocol corresponds to a local maximum point in the design space of parameters. To further improve the control performance of stochastic event-based protocols, an iterative parameter design algorithm (with finite iterations) is proposed for ensuring a strict reduction in cost functions after each iteration. Finally, numerical simulations are provided to illustrate the theoretical results.

Solving the Economic Growth Acceleration Model with Memory Effects: An Application of Combined Theorem of Adomian Decomposition Methods and Kashuri–Fundo Transformation Methods

The primary purpose of this study is to solve the economic growth acceleration model with memory effects for the quadratic cost function (Riccati fractional differential equation), using Combined Theorem of Adomian Polynomial Decomposition and Kashuri–Fundo Transformation methods. The economic growth model (EGM) with memory effects for the quadratic cost function is analysed by modifying the linear fractional differential equation. The study’s significant contribution is to develop a linear cost function in the EGM for a quadratic non-linear cost function and determine the specific conditions of the Riccati fractional differential equation (RFDEs) in the EGM with memory effects. The study results showed that RFDEs in the EGM involving the memory effect have a solution and singularity. Additionally, this study presents a comparison of exact solutions using Lie symmetry, Combined Theorem of Adomian Polynomial Decomposition, and Kashuri–Fundo Transformation methods. The results showed that the three methods have the same solution. Furthermore, this study provides a numerical solution to the RFDEs on the EGM with memory effects. The numerical simulation results showed that the output value of Y(t) for the quadratic cost function in the economic growth model is significantly affected by the memory effect.

Fault tolerant consensus control of multi-agent systems under dynamic event-triggered mechanisms

In this paper, the fault-tolerant consensus control (FTCC) problem is studied for multi-agent systems (MASs) with the dynamic event-triggered mechanism (DETM). To ease the communication burden, DETM governed by an additional internal dynamical variable is introduced. A novel fault-tolerant controller is presented to mitigate system performance degradation caused by failures, where a simple fault compensator is constructed through a protocol-based observer. Then, some sufficient conditions are established to examine the bounded consensus while optimizing the predetermined quadratic cost criterion. In addition, the explicit expression of the desired controller is also parameterized through orthogonal decomposition. Finally, two simulation results are made for the sake of verifying the effectiveness of the developed FTCC scheme, the fault compensation test as well as the quadratic cost one.

Pressure Sensor Placement for Leak Localization in Water Distribution Networks Using Information Theory.

This paper presents a method for optimal pressure sensor placement in water distribution networks using information theory. The criterion for selecting the network nodes where to place the pressure sensors was that they provide the most useful information for locating leaks in the network. Considering that the node pressures measured by the sensors can be correlated (mutual information), a subset of sensor nodes in the network was chosen. The relevance of information was maximized, and information redundancy was minimized simultaneously. The selection of the nodes where to place the sensors was performed on datasets of pressure changes caused by multiple leak scenarios, which were synthetically generated by simulation using the EPANET software application. In order to select the optimal subset of nodes, the candidate nodes were ranked using a heuristic algorithm with quadratic computational cost, which made it time-efficient compared to other sensor placement algorithms. The sensor placement algorithm was implemented in MATLAB and tested on the Hanoi network. It was verified by exhaustive analysis that the selected nodes were the best combination to place the sensors and detect leaks.

EFFECT OF RISK-SENSITIVE STOCHASTIC OPTIMAL CONTROL WITH TRACKING IN AN EVAPORATOR

This work presents an application of the Risk-Sensitive (R-S) control with tracking applied to a stochastic nonlinear system which models the operation of an electronic expansion valve (EEV) in a conventional evaporator. A novel dynamical stochastic equation represents the mathematical model of the evaporator system. The R-S stochastic optimal problem consists of the design of an optimal control u(t) such that the state reaches setpoint values (SP) and minimizes the exponential quadratic cost function. The presence of disturbances and errors in the sensor measurements is represented by Gauss white noise in the state equation, with the coefficient v(e/(2?^2 )) . One novel characteristic in this proposal is that the coefficient of the control into the state equation contains the state term. The error and exponential quadratic cost function show that the R-S control has a better performance versus the classical PID (Proportional, Integral Derivative) control. Key Words: Optimal Risk-Sensitive control with tracking, modelling of the evaporator.