Bilinear Terms Research Articles

Retrofit Design of Hydrogen Network in Refineries: Mathematical Model and Global Optimization

The problem of retrofit design of refinery hydrogen networks is addressed in this work, using the mathematical superstructure optimization. The superstructure of retrofit hydrogen network design contains hydrogen using, producing, and purifying units; along with compressors to facilitate hydrogen distribution. The developed mathematical model is formulated as a mixed integer nonlinear programming model (MINLP), with the objective being minimum total annual cost. The nonlinearity in the model is because of the bilinear, posynomia, and linear fractional terms. A new heuristic method is presented which helps in assigning suction and discharge pressures for the newly retrofitted compressor. With such an assignment, the nonlinearity in the model is now only confined to bilinear terms. This bilinear MINLP model is solved to global optimality using the proposed global optimization algorithm. Tests on some literature examples show that the proposed algorithm can reach global solutions faster than some commercial ...

Adaptive filtering for the identification of bilinear forms

Bilinear systems are involved in many interesting applications, especially related to the approximation of nonlinear systems. In this context, the bilinear term is usually defined in terms of an input–output relation (i.e., with respect to the data). Recently, a different approach has been introduced, by defining the bilinear term with respect to the impulse responses of a spatiotemporal model, which resembles a multiple-input/single-output (MISO) system. Also, in this framework, the Wiener filter has been studied to address the identification problem of these bilinear forms. Since the Wiener filter may not be always very efficient or convenient to use in practice, we propose in this paper an adaptive filtering approach. Consequently, we develop and analyze some basic algorithms tailored for the identification of bilinear forms, i.e., least-mean-square (LMS), normalized LMS (NLMS), and recursive-least-squares (RLS). Simulations performed in the context of system identification (based on the MISO system approach) support the theoretical findings and indicate the appealing performance of these algorithms.

English

A stochastic affine evolution equation with bilinear noise term is studied, where the driving process is a real-valued fractional Brownian motion with Hurst parameter greater than 1/2. Stochastic integration is understood in the Skorokhod sense. The existence and uniqueness of weak solution is proved and some results on the large time dynamics are obtained.

On the local frame in nonlinear higher-spin equations

Properties of the resolution operator dloc∗ in higher-spin equations, that leads to local current interactions at the cubic order and minimally nonlocal higher-order corrections, are formulated in terms of the condition on the class of master fields of higher-spin theory that restricts both the dependence on the spinor Y , Z variables and on the contractions of indices between the constituent fields in bilinear terms. The Green function in the sector of zero-forms is found for the case of constituent fields carrying helicities of opposite signs. It is shown that the local resolution dloc∗ differs from the conventional De Rham resolution dZ∗ by a non-local shift.

An Electric Vehicle Routing Optimization Model With Hybrid Plug-In and Wireless Charging Systems

In order to address the inconvenience of having to stop to charge the battery of an electric vehicle, wireless on-road charging technology, also known as charge-while-driving, has garnered much attention in recent studies. However, wireless charging devices may have a higher charge cost than traditional plug-in charging devices. Therefore, a multi-objective route optimization model based on model predictive control is established in this paper to determine an optimal route for drivers to coordinate wireless and plug-in charging strategies. To reduce the complexity of the proposed model due to its bilinear terms, the Big-M approach is employed to exactly linearize the bilinear terms by introducing dummy variables and additional constraints, which leads to a mixed integer linear programming model that can be solved efficiently. Finally, two systems are tested, including a real-world road map in Xi’an city to demonstrate the effectiveness of the proposed model.

An integrated chance-constrained stochastic model for efficient and sustainable supplier selection and order allocation

Effective allocation of scarce resources across supply chain environments is an emerging issue, as enterprises face shortfalls in raw materials, human labour, budgetary resources, equipment, energy and capacity. We consider these related objectives in designing efficient and sustainable supply networks using a multi-objective mixed-integer non-linear programming (MINLP) model for efficient and sustainable supplier selection and order allocation with stochastic demand. Our approach considers sustainability dimensions including economic, environmental and social responsibility, but also seeks to design the most efficient supply network given constraints of the supply market. Enterprise efficiency is assessed using a bi-objective data envelopment analysis (DEA) whose inputs include raw materials, current expenses and labour force capacity. The resulting model is non-convex because of the presence of bilinear terms in DEA-related constraints, so we introduce a multi-stage solution procedure that first uses piecewise McCormick envelopes (PCM) to linearise the bilinear terms. Next, we introduce a set of valid inequalities in order to improve solution time of the problem whose dimension significantly increases after being linearised. We then exploit chance constrained programming approaches to deal with stochastic demand. Finally, a single aggregated objective function is derived using a fuzzy multi-objective programming approach. A manufacturing case study demonstrates the validity of the proposed approach, and its effectiveness in designing a supply network that addresses the ‘triple bottom line’ of people, profit and planet that comprises many sustainability initiatives in an efficient manner.

Global optimisation of multi-plant manganese alloy production

This paper studies the problem of multi-plant manganese alloy production. The problem consists of finding the optimal furnace feed of ores, fluxes, coke, and slag that yields output products which meet customer specifications, and to optimally decide the volume, composition, and allocation of the slag. To solve the problem, a nonlinear pooling problem formulation is presented upon which the bilinear terms are reformulated using the Multiparametric Disaggregation Technique (MDT). This enables global optimisation by means of commercial software for mixed integer linear programs. We demonstrate the model and solution approach through case studies from a Norwegian manganese alloy producer. The computational study shows that the model and proposed optimisation approach can solve problem sizes of up to ten furnaces to a small optimality gap, that global optimization approach with MDT scales well with larger, real problem instances, and that the model outperforms the current operational practice.

The analytical one-loop contributions to Higgs boson mass in the Supersymmetric Standard Model with vector-like particles

In this paper, we extend the Minimal Supersymmetric Standard Model (MSSM) with additional vector-like particles (VLPs) and compared the differences of mass spectrum and Feynman rules between the MSSM and MSSMV. We analytically calculate the one loop contributions to the Higgs boson mass from the fermions and sfermions in the on shell renormalization scheme. After discussing and numerically analyzing cases without bilinear terms and a case with a (partial) decoupling limit, we find: (i) the corrections depend on the mass splittings between quarks and squarks and between vector-like fermions and their sfermions; (ii) there exists the (partial) decoupling limit, where the VLPs decouple from the electroweak (EW) energy scale, even when one of the VLPs is light around the EW scale. The reason is that the contributions to Higgs mass can be suppressed by the (or partial) decoupling effects, which can make the EW phenomenology different from the MSSM. Moreover, we present some numerical analysis to understand these unique features.

Uniqueness of Nash equilibrium in continuous two-player weighted potential games

The literature results about existence of Nash equilibria in continuous potential games [15] exploit the property that any maximum point of the potential function is a Nash equilibrium of the game (the vice versa being not true) and those about uniqueness use strict concavity of the potential function. The following question arises: can we find sufficient conditions on the potential function which guarantee one and only one Nash equilibrium when such a function is not strictly concave and the existence of a maximum is not ensured? The paper positively answers this question for two-player weighted potential games when the strategy sets are (not necessarily finite dimensional) real Hilbert spaces. Illustrative examples in finite dimensional spaces are provided, together with an application in infinite dimensional ones where a weighted potential function with a bilinear common interaction term is involved.

Time-varying and state-dependent recovery rates in epidemiological models.

Differential equation models of infectious disease have undergone many theoretical extensions that are invaluable for the evaluation of disease spread. For instance, while one traditionally uses a bilinear term to describe the incidence rate of infection, physically more realistic generalizations exist to account for effects such as the saturation of infection. However, such theoretical extensions of recovery rates in differential equation models have only started to be developed. This is despite the fact that a constant rate often does not provide a good description of the dynamics of recovery and that the recovery rate is arguably as important as the incidence rate in governing the dynamics of a system. We provide a first-principles derivation of state-dependent and time-varying recovery rates in differential equation models of infectious disease. Through this derivation, we demonstrate how to obtain time-varying and state-dependent recovery rates based on the family of Pearson distributions and a power-law distribution, respectively. For recovery rates based on the family of Pearson distributions, we show that uncertainty in skewness, in comparison to other statistical moments, is at least two times more impactful on the sensitivity of predicting an epidemic's peak. In addition, using recovery rates based on a power-law distribution, we provide a procedure to obtain state-dependent recovery rates. For such state-dependent rates, we derive a natural connection between recovery rate parameters with the mean and standard deviation of a power-law distribution, illustrating the impact that standard deviation has on the shape of an epidemic wave.

Natural emergence of neutrino masses and dark matter from R-symmetry

We propose a supersymmetric extension of the Standard Model (SM) with a continuous global U(1)R symmetry. The R-charges of the SM fields are identified with that of their lepton numbers. As a result, both bilinear and trilinear ‘R-parity violating’ (RPV) terms could be present at the superpotential. However, R-symmetry is not an exact symmetry as it is broken by supergravity effects. Hence, sneutrinos acquire a small vacuum expectation value in this framework. However, a suitable choice of basis ensures that the bilinear RPV terms can be completely rotated away from the superpotential and the scalar potential. On the other hand, the trilinear terms play a very crucial role in generating neutrino masses and mixing at the tree level. This is noticeably different from the typical R-parity violating Minimal Supersymmetric Standard Model. Also, gravitino mass turns out to be the order parameter of R-breaking and is directly related to the neutrino mass. We show that such a gravitino, within the mass range 200 keV ≲ m3/2 ≲ 0.1 GeV can be an excellent dark matter candidate. Finally, we looked into the collider implications of our framework.

Combined FDG and CT Radiomics Features Predict FMISO Uptake in Head and Neck Cancer

18F-fluoromisonidazole (FMISO) is a hypoxia PET imaging agent that has been used in head and neck to alter management. However, FMISO is still an investigational agent that requires an institutional review board with an IND and therefore is not widely available. Having a surrogate biomarker based on an approved imaging technique that could indirectly identify hypoxia would enable hypoxia-based treatment modulation in a routine clinical setting. In this work we assessed whether a radiomics biomarker built from FDG PET and CT features could predict FMISO uptake on a per-lesion basis. One hundred head and neck cancer patients undergoing chemoradiotherapy treatment at a single institution between 2011 and 2016 received pre-treatment FDG and FMISO PET/CT. A total of 134 lesions delineated by physicians on the treatment planning CT and with a volume larger than 10 cc were analyzed. Forty-six lesions were held out as a validation set. The level of hypoxia was defined as the maximum tumor-to-blood ratio (TBRmax) of FMISO uptake measured at ∼150 minutes post injection. Approximately 200 radiomics features were extracted from FDG PET (first order and run length features only) and treatment planning CT (all orders, including run length), and used to produce a prediction of FMISO TBRmax. The most frequent features were selected by lasso using 100×10-fold cross validation (CV). The resulting features and their bilinear interaction terms were used again in a lasso regression model. The final combination of predictors was found by minimizing the mean square error on nested 10-fold CV. Performance was assessed by means of (i) R2, (ii) F-test P-value for a constant-model null hypothesis, and (iii) area under the curve (AUC) analysis, assuming a hypoxia threshold of TBRmaxthresh = 1.4. The mean FMISO TBRmax of the population was 1.82 (SD = 0.65). The resulting model contained two individual FDG PET features (median SUV and low gray-level run emphasis) and three interaction terms with other PET run-length and CT Haralick correlation features. In the training dataset (n = 88) the resulting prediction had an adjusted R2 = 0.42. For a hypoxia threshold of TBRmaxthresh = 1.4, the AUC was 0.86. In the validation dataset the prediction had an adjusted R2 of 0.17, with F-test P = 0.002. Assuming TBRmaxthresh = 1.4, the AUC for the validation dataset was 0.79, with an optimal operating point of FPR = 0.21 and TPR = 0.78. We found that FDG PET and CT provide complementary information that can be used to form a radiomics biomarker that correlates with tumor hypoxia as evaluated by FMISO PET. Further validation using multi-institutional datasets is required. Clinically, this is a promising approach to identify a subset of head and neck radiotherapy patients who may benefit from hypoxia-based dose modification.

An optimization approach for deriving upper and lower bounds of transportation network vulnerability under simultaneous disruptions of multiple links

This paper aims to develop an optimization approach for deriving the upper and lower bounds of transportation network vulnerability under simultaneous disruptions of multiple links without the need to evaluate all possible combinations as in the enumerative approach. Mathematically, we formulate the upper and lower bounds of network vulnerability asa binary integer bi-level program (BLP). The upper-level subprogram maximizes or minimizes the remaining network throughput under a given number of disrupted links, which corresponds to the upper and lower vulnerability bounds. The lower-level subprogram checks the connectivity of each origin-destination (O-D) pair under a network disruption scenario without path enumeration. Two alternative modeling approaches are provided for the lower-level subprogram: the virtual link capacity-based maximum flow problem formulation and the virtual link cost-based shortest path problem formulation. Computationally, the BLP model can be equivalently reformulated asa single-level mixed integer linear program by making use of the optimality conditions of the lower-level subprograms and linearization techniques for the complementarity conditions and bilinear terms. Numerical examples are also provided to systematically demonstrate the validity, capability, and flexibility of the proposed optimization model. The vulnerability envelope constructed by the upper and lower bounds is able to effectively consider all possible combinations without the need to perform a full network scan, thus avoiding the combinatorial complexity of enumerating multi-disruption scenarios. Using the vulnerability envelope asa network performance assessment tool, planners and managers can more cost-effectively plan for system protection against disruptions, and prioritize system improvements to minimize disruption risks with limited resources.

Manifestation of higher-order inter-granular exchange in magnetic recording media

Exchange coupling between magnetic grains is essential for maintaining the stability of stored information in magnetic recording media. Using an atomistic spin model, we have investigated the coupling between neighbouring magnetic grains where magnetic impurity atoms have migrated into the non-magnetic grain boundary. We find that when the impurity density is low, a biquadratic term in addition to the bilinear term is found to better describe the inter-granular exchange coupling. The temperature dependence of both terms is found to follow a power law behaviour with the biquadratic exchange constant decaying faster than the bilinear. For the increasing grain boundary thickness, the inter-granular exchange reduces and also decays more quickly with temperature. Further simulations of a grain at a bit boundary show an unexpected energy minimum for in-plane magnetisation. This feature is reproduced if the biquadratic exchange term is included.

A New LMI-Based Output Feedback Controller Design Method for Discrete-Time LPV Systems with Uncertain Parameters

This paper deals with observer-based stabilization for a class of Linear Parameter-Varying (LPV) systems in discrete-time case. A new LMI design method is proposed to design the observer-based controller gains. The main contribution consists in providing a new and convenient way to use the congruence principle to reduce the conservatism of some existing results in the literature. This use of congruence principle leads to some additional slack matrices as decision variables, which make disappear some bilinear terms. To the authors’ best knowledge, this is the first time the congruence principle is exploited in this way. The effectiveness and superiority of the proposed design techniques, compared to existing results in the literature, are demonstrated through two numerical examples.

A piecewise McCormick relaxation-based strategy for scheduling operations in a crude oil terminal

The petroleum supply chain can be divided into three segments: upstream, midstream and downstream. This work studies the scheduling of operations in a crude oil terminal within the midstream segment. The first challenge consists in deciding how the crude oil that arrives in vessels should be uploaded to the storage tanks. At the same time, the operations engineer must decide which storage tanks will feed the pipeline connected to the refinery in order to satisfy its demand. This work concerns the crude oil terminal of the national refinery of Uruguay. To schedule terminal operations, this work proposes an iterative two-step MILP-NLP algorithm based on piecewise McCormick relaxation and a domain-reduction strategy for handling bilinear terms. For small instances for which an optimal solution is known, the proposed strategy consistently finds optimal or near-optimal solutions. It also solves larger instances which are, in some cases, intractable by a global optimization solver.

On the Identification of Bilinear Forms With the Wiener Filter

In this letter, the identification problem of bilinear forms with the Wiener filter is addressed. The contribution is twofold. First, a different approach is introduced, by defining the bilinear term with respect to the impulse responses of a spatiotemporal model, in the context of multiple-input/single-output systems. Second, two versions of the Wiener filter (namely direct and iterative) are developed in this context. Moreover, the advantage of the iterative Wiener filter is outlined as compared to the direct solution. The results of the simulations, which are performed from a system identification perspective, support the theoretical findings.

Massive spinons inS=1/2spin chains: Spinon-pair operator representation

Spinons are among the generic excitations in one-dimensional spin systems, they can be massless or massive. The quantitative description of massive spinons poses a considerable challenge in spite of various variational approaches. We show that a representation in terms of hopping and Bogoliubov spinon processes, which we call "spinon-pair" operators, and their combination is possible. We refer to such a representation as second quantized form. Neglecting terms which change the number of spinons yields the variational results. Treating the bilinear and quartic terms by continuous unitary transformations leads to considerably improved results. Thus, we provide the proof-of-principle that systems displaying massive spinons as elementary excitations can be treated in second quantization based on spinon-pair representation.

Global Optimization of Nonlinear Blend-Scheduling Problems

Abstract The scheduling of gasoline-blending operations is an important problem in the oil refining industry. This problem not only exhibits the combinatorial nature that is intrinsic to scheduling problems, but also non-convex nonlinear behavior, due to the blending of various materials with different quality properties. In this work, a global optimization algorithm is proposed to solve a previously published continuous-time mixed-integer nonlinear scheduling model for gasoline blending. The model includes blend recipe optimization, the distribution problem, and several important operational features and constraints. The algorithm employs piecewise McCormick relaxation (PMCR) and normalized multiparametric disaggregation technique (NMDT) to compute estimates of the global optimum. These techniques partition the domain of one of the variables in a bilinear term and generate convex relaxations for each partition. By increasing the number of partitions and reducing the domain of the variables, the algorithm is able to refine the estimates of the global solution. The algorithm is compared to two commercial global solvers and two heuristic methods by solving four examples from the literature. Results show that the proposed global optimization algorithm performs on par with commercial solvers but is not as fast as heuristic approaches.

A globally linearly convergent method for pointwise quadratically supportable convex–concave saddle point problems

We study the Proximal Alternating Predictor–Corrector (PAPC) algorithm introduced recently by Drori, Sabach and Teboulle [8] to solve nonsmooth structured convex–concave saddle point problems consisting of the sum of a smooth convex function, a finite collection of nonsmooth convex functions and bilinear terms. We introduce the notion of pointwise quadratic supportability, which is a relaxation of a standard strong convexity assumption and allows us to show that the primal sequence is R-linearly convergent to an optimal solution and the primal-dual sequence is globally Q-linearly convergent. We illustrate the proposed method on total variation denoising problems and on locally adaptive estimation in signal/image deconvolution and denoising with multiresolution statistical constraints.