Bilinear Terms Research Articles

Taylor’s theorem: A new perspective for neural tensor networks

Neural tensor networks have been widely used in a large number of natural language processing tasks such as conversational sentiment analysis, named entity recognition and knowledge base completion. However, the mathematical explanation of neural tensor networks remains a challenging problem, due to the bilinear term. According to Taylor’s theorem, a kth order differentiable function can be approximated by a kth order Taylor polynomial around a given point. Therefore, we provide a mathematical explanation of neural tensor networks and also reveal the inner link between them and feedforward neural networks from the perspective of Taylor’s theorem. In addition, we unify two forms of neural tensor networks into a single framework and present factorization methods to make the neural tensor networks parameter-efficient. Experimental results bring some valuable insights into neural tensor networks.

Robust Assignment of Natural Frequencies and Antiresonances in Vibrating Systems through Dynamic Structural Modification

This paper proposes a novel method for the robust partial assignment of natural frequencies and antiresonances, together with the partial assignment of the related eigenvectors, in lightly damped linear vibrating systems. Dynamic structural modification is exploited to assign the eigenvalues, either of the system or of the adjoint system, together with their sensitivity with respect to some parameters of interest. To handle with constraints on the feasible modifications, the inverse eigenvalue problem is cast as a minimization problem and a solution method is proposed through homotopy optimization. Variables lifting for bilinear and trilinear terms, together with bilinear and double-McCormick’s constraints, are exploited to provide a convexification of the problem and to boost the attainment of the global optimum. The effectiveness of the proposed method is assessed through four numerical examples.

Joint Routing and Charging Problem of Multiple Electric Vehicles: A Fast Optimization Algorithm

Logistics has gained great attentions with the prosperous development of e-commerce, which is often seen as the classic optimal vehicle routing problem. Meanwhile, electric vehicle (EV) has been widely used in logistic fleet to curb the emission of green house gases in recent years. Thus, solving the optimization problem of joint routing and charging of multiple EVs is in a urgent need, whose objective function includes charging time, charging cost, EVs travel time, usage fees of EV and revenue from serving customers. This joint problem is formulated as a mixed integer programming (MIP) problem, which, however, is NP-hard due to integer restrictions and bilinear terms from the coupling between routing and charging decisions. The main contribution of this paper lies at proposing an efficient two-stage algorithm that can decompose the original MIP problem into two linear programming (LP) problems, by exploiting the exactness of LP relaxation and eliminating the coupled term. This algorithm can achieve a near-optimal solution in polynomial time. In addition, another variant algorithm is proposed based on the two-stage one, to further improve the quality of solution. Compared with the state-of-the-art algorithm, extensive simulations are implemented to validate the effectiveness of the proposed algorithm.

Multimode two-dimensional vibronic spectroscopy. II. Simulating and extracting vibronic coupling parameters from polarization-selective spectra.

Experimental demonstrations of polarization-selection two-dimensional Vibrational-Electronic (2D VE) and 2D Electronic-Vibrational (2D EV) spectroscopies aim to map the magnitudes and spatial orientations of coupled electronic and vibrational coordinates in complex systems. The realization of that goal depends on our ability to connect spectroscopic observables with molecular structural parameters. In this paper, we use a model Hamiltonian consisting of two anharmonically coupled vibrational modes in electronic ground and excited states with linear and bilinear vibronic coupling terms to simulate polarization-selective 2D EV and 2D VE spectra. We discuss the relationships between the linear vibronic coupling and two-dimensional Huang-Rhys parameters and between the bilinear vibronic coupling term and Duschinsky mixing. We develop a description of the vibronic transition dipoles and explore how the Hamiltonian parameters and non-Condon effects impact their amplitudes and orientations. Using simulated polarization-selective 2D EV and 2D VE spectra, we show how 2D peak positions, amplitudes, and anisotropy can be used to measure parameters of the vibronic Hamiltonian and non-Condon effects. This paper, along with the first in the series, provides the reader with a detailed description of reading, simulating, and analyzing multimode, polarization-selective 2D EV and 2D VE spectra with an emphasis on extracting vibronic coupling parameters from complex spectra.

Strategic Policymaking for Implementing Renewable Portfolio Standards: A Tri-Level Optimization Approach

Appropriately designed renewable support policies can play a leading role in promoting renewable expansions and contribute to low emission goals. Meanwhile, ill-designed policies may distort electricity markets, put power utilities and generation companies on an unlevel playing field and, in turn, cause inefficiencies. This paper proposes a framework to optimize policymaking for renewable energy sources, while incorporating conflicting interests and objectives of different stakeholders. We formulate a tri-level optimization problem where each level represents a different entity: a state regulator, a power utility and a wholesale electricity market. To solve this tri-level problem, we exploit optimality conditions and develop a modification of the Column-and-Cut Generation (C&CG) algorithm that generates cuts for bilinear terms. The case study based on the ISO New England 8-zone test system reveals different policy trade-offs that policymakers face under different decarbonization goals and implementation scenarios.

Fermion and meson mass generation in non-Hermitian Nambu–Jona-Lasinio models

We investigate the effects of non-Hermiticity on interacting fermionic systems. We do this by including non-Hermitian bilinear terms into the 3+1 dimensional Nambu--Jona-Lasinio (NJL) model. Two possible bilinear modifications give rise to $\mathcal{PT}$ symmetric theories; this happens when the standard NJL model is extended either by a pseudovector background field $ig \bar\psi\gamma_5 B_\mu \gamma^\mu \psi$ or by an antisymmetric-tensor background field $g \bar\psi F_{\mu\nu}\gamma^\mu \gamma^\nu \psi$. The three remaining bilinears are {\it anti}-$\mathcal{PT}$-symmetric in nature, $ig \bar\psi B_\mu \gamma^\mu \psi, ig\bar\psi \gamma_5 \psi$ and $ig\bar\psi {1}\psi$, so that the Hamiltonian then has no overall symmetry. The pseudovector $ig \bar\psi\gamma_5 B_\mu \gamma^\mu \psi$ and the vector $ig \bar\psi B_\mu \gamma^\mu \psi$ combinations, are, in addition, chirally symmetric. Thus, within this framework we are able to examine the effects that the various combinations of non-Hermiticity, $\mathcal{PT}$ symmetry, chiral symmetry and the two-body interactions of the NJL model have on the existence and dynamical generation of a real effective fermion mass (a feature which is absent in the corresponding modified massless free Dirac models) as well as on the masses of the composite particles, the pseudoscalar and scalar mesonic modes ($\pi$ and $\sigma$ mesons). Our findings demonstrate that $\mathcal{PT}$ symmetry is not necessary for real fermion mass solutions to exist, rather the two-body interactions of the NJL model supersede the non-Hermitian bilinear effects. The effects of chiral symmetry are evident most clearly in the meson modes, the pseudoscalar of which will always be Goldstone in nature if the system is chirally symmetric. Second solutions of the mesonic equations are also discussed.

Comparison of mixed-integer relaxations with linear and logarithmic partitioning schemes for quadratically constrained problems

Piecewise relaxations are powerful techniques to compute strong dual bounds for (mixed-integer) quadratically constrained problems and provide starting points that can be used by local nonlinear solvers to generate high-quality primal solutions. They work by first partitioning the domain of one of the variables in every quadratic or bilinear term and then selecting the optimal interval in the partition through binary variables. A variety of formulations have been proposed that can be distinguished based on how the number of binary variables scales with the number of intervals. In this paper, we compare linear and logarithmic partitioning schemes that can be derived from the piecewise McCormick relaxation (PCM) or the multiparametric disaggregation technique (MDT). Specifically, we propose MDT for a generic numeric representation system, showing that it becomes a linear partitioning scheme when considering a single digit and varying the basis, and a logarithmic partitioning scheme when fixing the basis and varying the number of digits. Through the solution of 25 benchmark instances for the pooling problem, considering relaxations for the p-, q- and tp-formulations, we show that it is better to rely on the linear scheme from PCM for a small number of intervals and on the logarithmic scheme from base-2 MDT when the number is large. The results also show that significantly stronger dual bounds can be obtained compared to commercial global optimization solvers BARON and GloMIQO.

Decomposing Loosely Coupled Mixed-Integer Programs for Optimal Microgrid Design

Microgrids are frequently employed in remote regions, in part because access to a larger electric grid is impossible, difficult, or compromises reliability and independence. Although small microgrids often employ spot generation, in which a diesel generator is attached directly to a load, microgrids that combine these individual loads and augment generators with photovoltaic cells and batteries as a distributed energy system are emerging as a safer, less costly alternative. We present a model that seeks the minimum-cost microgrid design and ideal dispatched power to support a small remote site for one year with hourly fidelity under a detailed battery model; this mixed-integer nonlinear program (MINLP) is intractable with commercial solvers but loosely coupled with respect to time. A mixed-integer linear program (MIP) approximates the model, and a partitioning scheme linearizes the bilinear terms. We introduce a novel policy for loosely coupled MIPs in which the system reverts to equivalent conditions at regular time intervals; this separates the problem into subproblems that we solve in parallel. We obtain solutions within 5% of optimality in at most six minutes across 14 MIP instances from the literature and solutions within 5% of optimality to the MINLP instances within 20 minutes.

New superstructure-based model for the globally optimal synthesis of refinery hydrogen networks

Abstract This work presents a new superstructure for refinery hydrogen networks synthesis. The superstructure eliminates the need to have the pressures of compressors as variables, while both dedicated and shared compressors can be included. As a result of considering the pressures as the parameters known a-priori, the constraint of compression power is expressed by using linear equation. Hydrogen networks synthesis is then formulated as a new mixed integer nonlinear model whose nonlinearities are only attributed to the bilinear and concave terms. The proposed model is a simple optimization problem that could be solved to global optimality. Four literature examples and an industrial case are tested for illustration purpose. The obtained solution results exhibit 3.1–4.5 % reduction in the total annualized cost compared with those of literature ones.

Coordinated scheduling of integrated power and gas grids in consideration of gas flow dynamics

Natural gas, hydrogen, methane, or their mixture play important roles in the energy systems, and the coordinated optimal scheduling for the integrated energy systems are requisite in consideration of complicated system characteristics. This paper focuses on mixed-integer second-order cone programming-based optimal scheduling for the electric grids with gas flow dynamics in consideration of AC power flow, unit commitment, and line switching. For the gas grids, differential continuity equations and differential momentum equations, representing gas flow dynamics, are discretized to a group of algebraic equations with the implicit trapezoidal rules. In the algebraic equations, nonconvex bilinear terms with integer variables, representing gas flow directions, are transformed into linear inequality constraints with McCormick envelopes, and quadratic terms are relaxed by the second-order cone (SOC) approach. For the electric grids, SOC relaxation-based AC power flow models are employed, and the line switching problem is modeled by the improved SOC relaxation approach with McCormick envelopes. The entire problem is established as a mixed-integer second-order cone programming model, and the impacts of gas flow dynamics on the feasible region compared to the steady-state region are discussed.

Quantum States in Templated Biological Processes

Living matter is characterized by its variegated potential energy landscape possessing a proneness to continually absorb externally supplied energy. This enables it to ascend from its momentary energy minimum state to one of its myriad barriers only to subsequently descend to a new minimum with a potentiality to perform new functions or processes, in the while exuding energy (mainly in the form of heat). As in studies of molecular intersystem crossing, the jumping processes are describable in terms of quantum states. In this work we derive the low energy quantum states for those three templated self-assembling processes, self-replication, metabolism and self-repair that are commonly regarded as distinguishing animate from inanimate substance. The outcome of each process is a new, long-living, stable molecular aggregate characterized by its specific conformation, comprising a host of micro-states associated with sub-conformations and patterned upon the template. The provenance of these newly-formed states is obtained here by a unified formalism for all three processes, based on a Hamiltonian, constructed in an abstract Hilbert-space framework, whose essences are bilinear coupling terms in the Hamiltonian between the template and the bath, as well as between the reactants and the bath. Treating these terms by second order perturbation, one finds in low lying quantum states an alignment between the template and the product, somewhat analogous to the Kramers-Anderson superexchange mechanism, with the bath replacing the bridging anion and by exploitation of the decohering due to the randomness of the bath. The idea underlying this work, recurrent in the biological literature and here expressed in a Physics, Hamiltonian framework, is the correlative unity of the whole biological system comprising multiple organs.

State Constrained Control-Affine Parabolic Problems II: Second Order Sufficient 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 sufficient conditions relying on the Goh transform.

A Construction of Strict Lyapunov Functions for A Bilinear Balancing Model

This paper focuses on a two-dimensional compartmental model which is linear except for bilinear terms of balancing kinetics. It is an equation of Lotka-Volterra type, but its structure is not symmetric like the predator-prey equation. This paper elucidates fundamental challenges in non-local stability analysis of the model. In particular, obstacles in constructing strict Lyapunov functions are discussed. A method consisting of tilting and semi-global construction is presented to overcome the challenges. Its benefit is highlighted from the viewpoint of input-to-state stability with respect to inflow perturbation.

AMMI and SREG analysis for protein content in Vigna unguiculata (L.) Walp

Identification of cowpea genotypes with high protein content for specific environments, based on the genotype-environment interaction, has a positive impact in places where access to protein for human consumption is deficient. The objective of the study was to analyze the protein content of 10 cowpea bean genotypes in five environments in the Caribbean Region of Colombia. The randomized complete block design with four replications at each site was used. The analysis of the genotype-environment interaction (GEI) was performed using the AMMI (additive main effects and multiplicative interaction) and SREG (regression in sites) models, in which the main effects of genotypes (G) + GEI are part of the bilinear term of the model. The AMMI and SREG models and their biplots were useful in the analysis and interpretation of the protein content of cowpea beans from experiments carried out in multiple environments. The AMMI model identified genotypes 1, 4 and 8 as those with the greatest adaptability and stability, and the Monteria (MO7B), Mahates (MA7B) and Cerete (CE7B) environments as the most favorable. The SREG model identified a potential mega-environment constituted by the PN7B, MA7B and CE7B environments, in which genotypes 1, 2 and 3 presented greater adaptability and stability, while genotype 8 showed specific adaptability in MO7B. In both models, genotypes 6, 7 and 10 showed absence of adaptability and stability in the studied environments.

A MILP-based clustering strategy for integrating the operational management of crude oil supply

In this paper, we present a MILP clustering formulation for tackling the operational management of crude oil supply (OMCOS) proposed by de Assis et al. (2019). The OMCOS consists of defining the scheduling of vessel trips between offshore platforms and a crude oil terminal, combined with the scheduling of operations in a terminal to supply crude oil to distillation columns. The benefits of using the clustering solution as a pre-step before solving the OMCOS are: (a) reduces the number of routes for vessels; (b) simplifies offloading and unloading operations; (c) imposes rules for crude mixtures in clusters of storage tanks that minimize property variations; and (d) produces bounds on crude properties inside storage tanks that are used to linearize bilinear terms in blending constraints. Through the combination of clusters and a MILP-NLP decomposition, near optimal solutions were obtained for a set of representative instances of OMCOS at a reduced computational cost.

Models and algorithms for the product pricing with single-minded customers requesting bundles

We analyze a product pricing problem with single-minded customers, each interested in buying a bundle of products, if the total price is less than or equal to their budget. The objective of the seller is to maximize the total revenue and we assume that supply is unlimited for all products. We contribute to a missing piece of literature by giving some mathematical formulations for this single-minded bundle pricing problem. We first present a mixed-integer nonlinear program with bilinear terms in the objective function and the constraints. By applying classical linearization techniques, we obtain two different mixed-integer linear reformulations. Inspired by a reformulation–linearization technique framework, we derive valid inequalities leading to a tighter linear reformulation. We then discuss a Benders decomposition to project strong cuts from the tightest model onto a light and fast model. In another attempt to exploit the tighter linear relaxation, we discuss heuristics based on the developed mixed-integer linear formulations to quickly find good solutions. We conclude this work with extensive numerical experiments to assess the quality of the mixed-integer linear formulations, as well as the performance of the Benders decomposition and the heuristics.

Energetic requirements of the transition from solitary to group living

Synergy is known to be vital for the group collaboration among non-kin individuals. In order to evaluate the condition of synergy that initiates group living, we build a model of food intake based on three types of functional response. We show that type III functional response is prerequisite for synergy to allow group living. The optimal number of gathering individuals can be also evaluated from Type III functional response curve. Type III functional response consists of terms depending linearly and bilinearly on the number of individuals and the bilinear term represents synergy. For a fixed value of the linear coefficient, there are upper and lower boundaries of the bilinear coefficient for synergistic collaboration. The dilution effect can be incorporated into the model through the functional response of the predator. Thus, the functional response of the predator as well as that of the prey contribute to the group living of the prey. Our model shows that group livings can be categorized into three types, namely those due to (1) synergy effect of the own group, (2) dilution effect against predators, and (3) both effects contributing together. The predator's functional response plays a decisive role in the last two types, where the predator response should be of anti-Type III (i.e., Type II).

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.