Bilinear Terms Research Articles

Multiobjective Endmember Extraction Based on Bilinear Mixture Model

Hyperspectral imagery is always composed of mixed pixels because of the limited spatial resolution of a sensor and the macroscopic/microscopic mixture of distinct substances. The linear mixing model (LMM) is proven to be simple and effective in extensive literature when the macroscopic mixture dominates the mixing process. But when the photons undergo multiple reflections before reaching the sensor, the LMM becomes invalid. In this circumstance, the bilinear mixture model (Bi-LMM), which considers secondary reflections with a bilinear term, is a viable alternative. However, the bilinear term in most existing Bi-LMMs is constructed based on the pre-estimated endmembers, and thus, most Bi-LMMs focus mainly on the abundance estimation. This may lead to inaccurate estimation of endmembers and abundances for a given hyperspectral image. In this article, we propose a multiobjective endmember extraction (Bi-MoEE) method within the bilinear mixture paradigm, which considers each secondary reflection as a virtual endmember. Then, Bi-MoEE selects real and virtual endmembers from an extended spectral library consisting of a standard spectral library and their virtual products. By imposing some intuitive constraints, the solution space is greatly reduced, and the multipoint crossover and restricted bit-flip mutation operators are specially designed. Finally, Bi-MoEE can efficiently obtain a set of tradeoff solutions by minimizing the unmixing residuals and the number of selected endmembers, and automatically determine the optimal solution with multiobjective decision-making techniques. Compared with some advanced endmember extraction methods, the proposed Bi-MoEE does not need to know the number of real endmembers. In addition, the time efficiency of Bi-MoEE is mainly related to the image size and the algorithmic parameters, and has little to do with the size of spectral library, thus facilitating the practical implementation of Bi-MoEE with regard to the oversized spectral library. The experiments on synthetic and real data sets demonstrated the excellent performance of Bi-MoEE.

A Historical-Correlation-Driven Robust Optimization Approach for Microgrid Dispatch

Robust optimization (RO) has been a proficient approach for uncertainty dispatch in microgrids. In view of the strong conservativeness of traditional RO models, this article proposes a historical-correlation-driven (HCD) RO method for the MG dispatch problem. Firstly, the distinct spatiotemporal correlations in renewable energy (RE) generation are confirmed based on the historical data series of RE power. To effectively extract the correlation characteristics, the new intervals that envelop all correlation data points of similar days are formulated via the line-fitting approach, and the interval boundaries are added as liner constraints into the polyhedral uncertainty set that actively integrates the spatiotemporal correlations and avoids unreasonable scenarios. Secondly, a gradient-approximating decomposition algorithm with equilibrium constraints is put forward to address the nonlinear RO problem caused by non-independent uncertainties. The alternative optimization procedure ensures the gradient ascent of the objective, thus achieving the fast solution convergence of the Max-Min model with binary recourse variables. Besides, the equilibrium constraints realize the precise linearization of bilinear terms, as well as avoid the introduction of numerous binaries. Case tests verify the effectiveness and superiority of the proposed HCD-RO method comparing with existing models and algorithms.

State-Constrained Control-Affine Parabolic Problems I: First and Second Order Necessary Optimality Conditions

In this paper we consider an optimal control problem governed by a semilinear heat equation with bilinear control-state terms and subject to control and state constraints. The state constraints are of integral type, the integral being with respect to the space variable. The control is multidimensional. The cost functional is of a tracking type and contains a linear term in the control variables. We derive second order necessary conditions relying on the concept of alternative costates and quasi-radial critical directions. The appendix provides an example illustrating the applicability of our results.

Model predictive control design for constrained Markov jump bilinear stochastic systems with an application in finance

In this study, we propose a solution to the model predictive control problem for a class of constrained discrete-time bilinear stochastic systems consisting of two coupled subsystems with Markov jumps. The first one includes a bilinear term in the state variables of the second subsystem and the input, whereas the second subsystem is described by a Markov switching vector autoregressive model. Furthermore, hard constraints imposed on the input manipulated variables. The results obtained are applied to the dynamic investment portfolio selection problem for a financial market with serially dependent returns and switching modes, subject to hard constraints on trading amounts. Our approach is tested on a real dataset from the New York Stock Exchange and the Russian Stock Exchange MOEX.

A generalized benders decomposition-based global optimization approach to symbolic regression for explicit surrogate modeling from limited data information

Solving noncovex Disjunctive Programming (DP) problems for representing symbolic regression trees to build explicit surrogate models from limited data information is challenging. This paper presents an effective global optimization approach to deal with these DPs. Piecewise McCormick envelope-based strategy is used to relax the nonconvex bilinear terms and nonconvex inequality constraints of the DP. A large-scale convex mixed integer nonlinear programing (MINLP) is then generated via the convex hull. A tailored linearization strategy further relaxes tightly all nonlinear terms in the convex MINLP to generate a mixed integer linear programming which is finally solved to global optimality by Generalized Benders Decomposition-based algorithm. Through testing three numerical simulations and Cetane number prediction, when compared to the monolithic approach of the state-of-the-art global optimization solvers such as BARON, the proposed approach is shown to reduce the solution time by up to two orders of magnitude and construct more accurate explicit surrogate models.

Multi‐stage risk‐based dispatch considering short‐term reliability of transmission lines

The reliability of transmission lines (TLs) has great influences on the safe operation of power systems. Random failures from TLs can be caused by multiple uncertainties such as the high load current. In this study, a multi-stage risk-based dispatch model is proposed to minimise the sum of generation cost and risk cost. Impacts of natural ageing, load current, health condition and disaster weather on the short-term reliability of TLs are studied. For ensuring the operation security of power systems in the post-contingency stage, the dynamic thermal rating (DTR) technology is implemented to improve capacities of TLs for coping with the potential overload. In order to solve this non-linear optimiation problem caused by load current-dependent failure probabilities and thermal rating constraints, a double-iteration solving strategy is proposed. In exterior iterations, the sequential linear programming is applied to decouple bilinear terms of the risk cost and locally linearise failure probabilities. Meanwhile, in interior iterations, the Benders decomposition is utilised to further divide the model into the main problem and the sub-problem for coordinating the preventive and corrective control and checking the feasibility of DTR, respectively. Proposed approaches are validated on a modified IEEE 24-bus test system.

Finite temperature effects in quantum systems with competing scalar orders

The study of the competition or coexistence of different ground states in many-body systems is an exciting and actual topic of research, both experimentally and theoretically. Quantum fluctuations of a given phase can suppress or enhance another phase depending on the nature of the coupling between the order parameters, their dynamics and the dimensionality of the system. The zero temperature phase diagrams of systems with competing scalar order parameters with quartic and bilinear coupling terms have been previously studied for the cases of a zero temperature bicritical point and of coexisting orders. In this work, we apply the Matsubara summation technique from finite temperature quantum field theory to introduce the effects of thermal fluctuations on the effective potential of these systems. This is essential to make contact with experiments. We consider two and three-dimensional materials characterized by a Lorentz invariant quantum critical theory, i.e., with dynamic critical exponent z = 1, such that time and space scale in the same way. We obtain that in both cases, thermal fluctuations lead to weak first-order temperature phase transitions, at which coexisting phases arising from quantum corrections become unstable. We show that above this critical temperature (Tc), the system presents scaling behavior consistent with that approaching a quantum critical point. Below the transition the specific heat has a thermally activated contribution with a gap related to the size of the domains of the ordered phases. We obtain that Tc decreases as a function of the distance to the zero temperature classical bicritical point (ZTCBP) in the coexistence region, implying that in our approach, the system attains the highest Tc above the fine tuned value of this ZTCBP.

Workload Transfer Strategy of Urban Neighboring Data Centers With Market Power in Local Electricity Market

This paper investigates how to minimize the operational cost of cloud service provider (CSP) that operates urban neighboring data centers (DCs) in the same electricity market and can conduct workload transfer among DCs. Due to the substantial electricity demand of DCs, their market power which can have impact on the locational marginal prices (LMPs) of the electricity market should be taken into consideration. We formulate a bilinear bilevel problem which regards the CSP as a price maker and explores cost-minimizing workload transfer strategies. The upper level is the operational cost minimization problem of CSP and the lower level corresponds to the economic dispatch problem of independent system operator (ISO) of electricity market which determines the electricity prices. It is challenging to directly solve the bilevel problem with bilinear term in the objective function. Hence, we first reformulate the original problem into a single level problem and then based on the property of the problem we develop a polytope cutting algorithm that attains the global optimal solution. The proposed algorithm solves linear optimizations iteratively by cutting the non-convex polytope feasible set into convex sets. In addition, considering the varying communication environment in practice, we analyze the impact of transfer price uncertainty on total cost of CSP, and show that the expected cost surprisingly decreases with the increasing uncertainty. Simulations based on the standard IEEE test cases show that the cost of CSP is significantly reduced and a win-win result for both the CSP and independent system operator (ISO) is possible.

Bilinear Minimax Optimal Control Problems for a von Kárman System with Long Memory

In the present article, we consider a von Kárman equation with long memory. The goal is to study a quadratic cost minimax optimal control problems for the control system governed by the equation. First, we show that the solution map is continuous under a weak assumption on the data. Then, we formulate the minimax optimal control problem. We show the first and twice Fréchet differentiabilities of the nonlinear solution map from a bilinear input term to the weak solution of the equation. With the Fréchet differentiabilities of the control to solution mapping, we prove the uniqueness and existence of an optimal pair and find its necessary optimality condition.

Unique weak solutions of the magnetohydrodynamic equations with fractional dissipation

AbstractThis paper examines the existence and uniqueness of weak solutions to the d‐dimensional magnetohydrodynamic (MHD) equations with fractional dissipation and fractional magnetic diffusion . The aim is at the uniqueness of weak solutions in the weakest possible inhomogeneous Besov spaces. We establish the local existence and uniqueness in the functional setting and when , and . The case when with and has previously been studied in [7, 19]. However, their approaches can not be directly extended to the fractional case when due to the breakdown of a bilinear estimate. By decomposing the bilinear term into different frequencies, we are able to obtain a suitable upper bound on the bilinear term for , which allows us to close the estimates in the aforementioned Besov spaces.

Non-Hermitian extension of the Nambu–Jona-Lasinio model in 3+1 and 1+1 dimensions

This paper presents a non-Hermitian PT-symmetric extension of the Nambu--Jona-Lasinio (NJL) model of quantum chromodynamics in 3+1 and 1+1 dimensions. In 3+1 dimensions, the SU(2)-symmetric NJL Hamiltonian $H_{\textrm{NJL}} = \bar\psi (-i \gamma^k \partial_k + m_0) \psi - G [ (\bar\psi \psi)^2 + (\bar\psi i \gamma_5 \vec{\tau} \psi)^2 ]$ is extended by the non-Hermitian, PT- and chiral-symmetric bilinear term $ig\bar\psi \gamma_5 B_{\mu} \gamma^{\mu} \psi$; in 1+1 dimensions, where $H_{\textrm{NJL}}$ is a form of the Gross-Neveu model, it is extended by the non-Hermitian PT-symmetric but chiral symmetry breaking term $g \bar\psi \gamma_5 \psi$. In each case, the gap equation is derived and the effects of the non-Hermitian terms on the generated mass are studied. We have several findings: in previous calculations for the free Dirac equation modified to include non-Hermitian bilinear terms, contrary to expectation, no real mass spectrum can be obtained in the chiral limit; in these cases a nonzero bare fermion mass is essential for the realization of PT symmetry in the unbroken regime. Here, in the NJL model, in which four-point interactions are present, we {\it do} find real values for the mass spectrum also in the limit of vanishing bare masses in both 3+1 and 1+1 dimensions, at least for certain specific values of the non-Hermitian couplings $g$. Thus, the four-point interaction overrides the effects leading to PT symmetry-breaking for these parameter values. Further, we find that in both cases, in 3+1 and in 1+1 dimensions, the inclusion of a non-Hermitian bilinear term can contribute to the generated mass. In both models, this contribution can be tuned to be small; we thus fix the fermion mass to its value when $m_0=0$ in the absence of the non-Hermitian term, and then determine the value of the coupling required so as to generate a bare fermion mass.

Tight convex relaxation for TEP problem: a multiparametric disaggregation approach

In recent years, there has been an increasing interest in using AC power flow equations for the transmission expansion planning (TEP) studies. The AC power flow equations are quadratic and hence the TEP problem can be formulated as a mixed-integer quadratically constrained programme. Therefore, the complexity of the TEP problem lies in the non-convexity of AC power flow equations in which the global optimal solution is not guaranteed to be found. This study aims at proposing a tight convex relaxation for the TEP problem. In this context, first, the TEP problem is formulated as a mixed-integer bilinear problem by representing the complex bus voltage in its rectangular coordinates. Second, the multiparametric disaggregation technique (MDT) and piecewise McCormick relaxation are employed to generate a mixed-integer linear relaxation. MDT is based on the discretisation of the domain of one of the variables in every bilinear term. The method presented is much more precise compared with the DC or other linearisation approaches, while the optimal solution is of high quality. The results of the case studies show the tractability and exactness of the proposed model as well as its superiority over the state-of-the-art schemes.

Hyperspectral image denoising by total variation-regularized bilinear factorization

Hyperspectral image (HSI) denoising is a prevalent research topic in the remote sensing area. In general, HSIs are inevitably impaired by different types of noise during the data acquisition. To fully characterize the underlying structures of clean HSI and remove mixed noises, we introduce a novel HSI denoising method named total variation-regularized bilinear factorization (BFTV) model. Specifically, we first utilize the bilinear factorization term to explore the globally low-rank structure of the clean HSI and suppress a certain degree of Gaussian noise, so as to make BFTV free to the singular value decomposition. Then the l1-norm is applied to detect and separate the mixed sparse noise including impulse noise, deadlines, and stripes. Besides, the TV regularization is introduced to describe the locally piecewise smoothness property of the clean HSI both in spatial and spectral domains. To solve this optimization problem, we devise an effective algorithm based on the augmented Lagrange multiplier method. Numerical experiments on five different kinds of mixed noise scenarios and one real world data have tested and demonstrated the superior denoising power of the proposed BFTV model compared with three state-of-the-art low-rank-based approaches.

Lbl Lstm Log Bilinear And Long Short Term Memory Based Efficient Stock Forecasting Model Considering External Fluctuating Factor

Stock market prediction problem is considered to be NP-hard problem because of highly volatile nature of stock market. In this paper, effort has been made to design efficient stock forecasting model using log Bilinear and long short term memory (LBL-LSTM) considering external fluctuating factor such as varying Bank's lending rates. The external factor bank's lending rates affects stock market performance ,as it plays vital role for the purchase of stocks in case of financial crisis faced by various business enterprises. Proposed LBL-LSTM based model shows performance improvement over existing machine learning algorithms used for stock market prediction.

Optimal design and operation of refinery hydrogen systems under multi-scale uncertainties

Robust design of hydrogen systems for large-scale refineries under multi-scale uncertainties has great potential for energy saving and cost reduction. The operation of hydrogen network is greatly affected by the rapidly changing market, deteriorating crude feedstock and varying operating conditions. In this paper, we propose a two-stage stochastic programming with flexibility constraints modeling framework that accounts for multi-scale uncertainties in an integrated strategy, aiming to achieve a good trade-off between the economic performance and the robustness of operation. A tailored-made global optimization algorithm based on multiparametric disaggregation technique is developed to tackle the computational challenge of the resulting MINLP problem with numerous bilinear terms. To illustrate the applicability of the proposed modeling framework and solution algorithm, studies on the optimal design of an industrial hydrogen system are presented. Multi-scenario operation strategy increases the operational flexibility and greatly reduces the total annualized cost in terms of the value of stochastic solution.

Preprocessing algorithm and tightening constraints for multiperiod blend scheduling: cost minimization

While a range of models have been proposed for the multiperiod blend scheduling problem (MBSP), solving even medium-size MBSP instances remains challenging due to the presence of bilinear terms and binary variables. To address this challenge, we develop solution methods for MBSP focusing on the cost minimization objective. We develop a novel preprocessing algorithm to calculate lower bounds on stream flows. We define product dedicated flow variables to address product specific features involved in MBSP. Bounds on stream flows and new product dedicated flow variables are then used to generate tightening constraints which significantly improve the solution time of the mixed integer nonlinear programming models as well as models based on linear approximations.

Optimal governmental incentives for biomass cofiring to reduce emissions in the short-term

Several studies have shown that biomass cofiring is a viable short-term option for coal-fired power plants to reduce their emissions if supported by appropriate tax incentives. These results suggest a unique opportunity for governments to design monetary incentives such as tax credits that lead to reduction in greenhouse gas emissions in biomass-rich regions. Therefore, a natural question is: What is an optimal tax credit strategy in these regions? To this end, we propose a Stackelberg/Nash game, and solve it algorithmically via reformulating the model as a mixed-integer bilinear program and using a piecewise linear relaxation of bilinear terms. The structure of the optimal solution of special cases is exploited, which helps design efficient heuristics. This study develops a case study using real data about power plants and biomass availability in Mississippi and Arkansas. The results compare the optimal tax credit schemes and plants’ cofiring strategies to provide insights on optimal tax credit mechanisms. Results show that a flexible tax credit scheme, which allows a plant-specific tax credit rate, is more efficient than the currently-used flat tax credit rate. This proposed approach uses a smaller budget and targets the plants that need funding support to comply with emissions regulations.

Natural Gas Flow Solvers Using Convex Relaxation

The vast infrastructure development, gas flow (GF) dynamics, and complex interdependence of gas with electric power networks call for advanced computational tools. Solving the equations relating gas injections to pressures and pipeline flows lies at the heart of natural gas network (NGN) operation, yet existing solvers that require careful initialization and uniqueness has been an open question. In this context, this article considers the nonlinear steady-state version of the GF problem. It first establishes that the solution to the GF problem is unique under arbitrary NGN topologies, compressor types, and sets of specifications. For GF setups where pressure is specified on a single (reference) node and compressors do not appear in cycles, the GF task is posed as n convex minimization. To handle more general setups, a GF solver relying on a mixed-integer quadratically constrained quadratic program (MI-QCQP) is also devised. This solver can be used for any GF setup at any NGN. It introduces binary variables to capture flow directions, relaxes the pressure drop equations to quadratic inequality constraints, and uses a carefully selected objective to promote the exactness of this relaxation. The relaxation is probably exact in NGNs with nonoverlapping cycles and a single fixed-pressure node. The solver handles efficiently the involved bilinear terms through McCormick linearization. Numerical tests validate our claims, demonstrate that the MI-QCQP solver scales well, and that the relaxation is exact even when the sufficient conditions are violated, such as in NGNs with overlapping cycles and multiple fixed-pressure nodes.

A multi-objective reactive distillation optimization model for Fischer–Tropsch synthesis

In the design of a reactive distillation column, aspects such as column configuration, catalyst loading, tray temperature, and side extraction rates should be well considered. Though preferences in Fischer–Tropsch (FT) synthesis may vary, it is acknowledged that the final product contains a wide range of hydrocarbons including fuel gas, gasoline, diesel, and linear wax. Zhang et al. (2018) previously developed an equation-oriented framework for optimal synthesis of integrated reactive distillation systems for FT processes.Here, we extend the mass, equilibrium, summation, and heat equations to a mathematical program with complementarity constraints to deal with possible dry trays in the non-reactive sections. The purpose of describing disappearing phases is to avoid infeasibilities due to multiple bilinear terms in the model for complicated model structures. The model is implemented by solving initialization steps and a sequence of nonlinear programming problems to determine an optimal structure and operating conditions.Design specifications for multiple products could be set as individual objectives to determine design limits. Moreover, a balance of multiple objectives could be reached by formulating the reactive distillation model as a multi-objective optimization problem. In this work, we employ the augmented ϵ-constraint method. The results show that significant design insights can be gained from the Pareto-optimal front regarding acceptable trade-offs among various objectives.

Error estimates for the finite element approximation of bilinear boundary control problems

In this article a special class of nonlinear optimal control problems involving a bilinear term in the boundary condition is studied. These kind of problems arise for instance in the identification of an unknown space-dependent Robin coefficient from a given measurement of the state, or when the Robin coefficient can be controlled in order to reach a desired state. Necessary and sufficient optimality conditions are derived and several discretization approaches for the numerical solution of the optimal control problem are investigated. Considered are both a full discretization and the postprocessing approach meaning that we compute an improved control by a pointwise evaluation of the first-order optimality condition. For both approaches finite element error estimates are shown and the validity of these results is confirmed by numerical experiments.