Bilinear Terms Research Articles

Existence of Néel Order in the S=1 Bilinear-Biquadratic Heisenberg Model via Random Loops

We consider the general spin-1 SU(2) invariant Heisenberg model with a two-body interaction. A random loop model is introduced and relations to quantum spin systems is proved. Using this relation it is shown that for dimensions 3 and above N\'eel order occurs for a large range of values of the relative strength of the bilinear ($-J_1$) and biquadratic ($-J_2$) interaction terms. The proof uses the method of reflection positivity and infrared bounds. Links between spin correlations and loop correlations are proved.

Optimal Design of Vibrating Systems Through Partial Eigenstructure Assignment

This paper proposes an inverse structural modification method for eigenstructure assignment (EA), which allows to assign the desired mode shapes only at the parts of interest of the system. The presence of unimposed eigenvector entries leads to a nonconvex problem. Therefore, to boost the convergence to a global optimal solution, a homotopy optimization strategy is implemented based on the convex approximation of the cost function. Such a relaxation is performed through some auxiliary variables and through the McCormick's relaxation of the occurring bilinear terms. The approach handles general assignment tasks, with an arbitrary number of modification parameters and prescribed eigenpairs.

Controller Design Based On Fuzzy Observers For T-S Fuzzy Bilinear Models

This article is devoted to the design of a fuzzy-observer-based control for a class of nonlinear systems with bilinear terms. The class of systems considered is the Takagi-Sugeno (T-S) fuzzy bilinear model. A new procedure to design the observer-based fuzzy controller for this class of systems is proposed. The aim is to design the fuzzy controller and the fuzzy observer of the augmented system separately in order to guarantee that the error between the state and its estimation converges faster to zero. By using the Lyapunov function, sufficient conditions are derived such that the closed-loop system is globally asymptotically stable. Moreover, sufficient conditions for controller design based on state estimation for robust stabilization of TS FBS with parametric uncertainties is proposed. The observer and controller gains are obtained by solving some linear matrix inequalities (LMIs). An example is given to illustrate the effectiveness of the proposed approach.

Unknown input observer design for one-sided Lipschitz discrete-time systems subject to time-delay

In this paper, we address the problem of unknown input observer design, which simultaneously estimates state and unknown input, of a class of nonlinear discrete-time systems with time-delay. A novel approach to the state estimation problem of nonlinear systems where the nonlinearities satisfy the one-sided Lipschitz and quadratically inner-bounded conditions is proposed. This approach also allows us to reconstruct the unknown inputs of the systems. The nonlinear system is first transformed to a new system which can be decomposed into unknown-input-free and unknown-input-dependent subsystems. The estimation problem is then reduced to designing observer for the unknown-input-free subsystem. Rather than full-order observer design, in this paper, we propose observer design of reduced-order which is more practical and cost effective. By utilizing several mathematical techniques, the time-delay issue as well as the bilinear terms, which often emerge when designing observers for nonlinear discrete-time systems, are handled and less conservative observer synthesis conditions are derived in the linear matrix inequalities form. Two numerical examples are given to show the efficiency and high performance of our results.

Incorporating Price-Responsive Demand in Energy Scheduling Based on Consumer Payment Minimization

In the restructured power industry, demand responsiveness is a key factor whose importance will be boosted due to the impetus provided by the development of smart grids. Within the context of pool-based electricity markets, this paper addresses the incorporation of price-responsive demand in multiperiod energy scheduling driven by consumer payment minimization. Although consumer payment minimization has drawn considerable attention mainly in an operational setting and also recently under a planning framework, available models and solution approaches typically neglect demand-side participation. The proposed scheduling model considers a marginal pricing scheme as well as the effects of both network constraints and intertemporal constraints associated with generation operation. Modeling demand-side participation leads to bilinear payment terms that significantly increase the mathematical complexity of the optimization process. The resulting problem is formulated as a mixed-integer nonlinear bilevel program for which no exact solution technique is currently available. This paper presents a novel methodology by which the original bilevel and bilinear problem is converted into an equivalent single-level mixed-integer linear program suitable for efficient off-the-shelf software. This transformation is based on the application of duality theory of linear programming, integer algebra, and Karush–Kuhn–Tucker optimality conditions. The proposed approach has been successfully applied to the IEEE 118-bus system.

Matrix Factorizations based on induced norms

We decompose a matrix Y into a sum of bilinear terms in a stepwise manner, by considering Y as a mapping between two finite dimensional Banach spaces. We provide transition formulas, and represent them in a duality diagram, thus generalizing the well known duality diagram in the french school of data analysis. As an application, we introduce a family of Euclidean multidimensional scaling models.

Existence of global strong solution for Korteweg system with large infinite energy initial data

This work is devoted to the study of the initial boundary value problem for a general isothermal model of capillary fluids derived by J.E. Dunn and J. Serrin (1985) (see [18]), which can be used as a phase transition model. We aim at proving the existence of local and global (under a condition of smallness on the initial data) strong solutions with initial density ln⁡ρ0 belonging to the Besov space B2,∞N2. It implies in particular that some classes of discontinuous initial density generate strong solutions. The proof relies on the fact that the density can be written as the sum of the solution ρL of the associated linear system and a remainder term ρ¯; this last term is more regular than ρL provided that we have regularizing effects induced on the bilinear convection term. The main difficulty consists in obtaining new estimates of maximum principle type for the associated linear system; this is based on a characterization of the Besov space in terms of the semi-group associated with this linear system. We show in particular the existence of global strong solution for small initial data in (B˜2,∞N2−1,N2∩L∞)×B2,∞N2−1; it allows us to exhibit a family of large energy initial data when N=2 providing global strong solution. In conclusion we introduce the notion of quasi-solutions for the Korteweg's system (a tool which has been developed in the framework of the compressible Navier–Stokes equations [31,30,32,26,27]) which enables to obtain the existence of global strong solution with a smallness condition which is subcritical. Indeed we can deal with large initial velocity in B2,1N2−1. As a corollary, we get global strong solution for highly compressible Korteweg system when N≥2. It means that for any large initial data (under an irrotational condition on the initial velocity) we have the existence of global strong solution provided that the Mach number is sufficiently large.

The Korteweg–de Vries equation on the half-line

The initial-boundary value problem (ibvp) for the Korteweg–de Vries (KdV) equation on the half-line with data in Sobolev spaces is analysed by combining the unified transform method with a contraction mapping approach. First, the linear KdV ibvp with initial and boundary data in Sobolev spaces is solved and the basic space and time estimates of the solution are derived. Then, further linear estimates in a new norm, motivated by the KdV bilinear term, are obtained. Finally, well-posedness of the KdV ibvp with data (u(x, 0), u(0, t)) in , , is established via a fixed point argument in an appropriate solution space.

Divisive heuristic for modularity density maximization

In this paper we consider a particular method of clustering for graphs, namely the modularity density maximization. We propose a hierarchical divisive heuristic which works by splitting recursively a cluster into two new clusters by maximizing the modularity density, and we derive four reformulations for the mathematical programming model used to obtain the optimal splitting. We report computational results of the eight algorithms (four reformulations with two different symmetry breaking strategies) obtained on some instances from the literature. Statistical tests show that the best model in terms of computational time is the one that is obtained with a dual reformulation of the bilinear terms arising in the objective function. Moreover, the hierarchical divisive heuristic provides generally near-optimal solutions in terms of modularity density.

Observer Design for One-Sided Lipschitz Discrete-Time Systems Subject to Delays and Unknown Inputs

In this paper, we address the problem of observer design for a class of nonlinear discrete-time systems in the presence of delays and unknown inputs. The nonlinearities studied in this work satisfy the one-sided Lipschitz and quadratically inner-bounded conditions which are more general than the traditional Lipschitz conditions. Both $\mathcal{H}_{\infty}$ observer design and asymptotic observer design with reduced-order are considered. The designs are novel compared to other relevant nonlinear observer designs subject to time delays and disturbances in the literature. In order to deal with the time-delay issue as well as the bilinear terms which usually appear in the problem of designing observers for discrete-time systems, several mathematical techniques are utilized to deduce observer synthesis conditions in the linear matrix inequalities form. A numerical example is given to demonstrate the effectiveness and high performance of our results.

Criticality and Chaos in Systems of Communities

We consider a simple model of communities interacting via bilinear terms. After analyzing the thermal equilibrium case, which can be described by an Hamiltonian, we introduce the dynamics that, for Ising-like variables, reduces to a Glauber-like dynamics. We analyze and compare four different versions of the dynamics: flow (differential equations), map (discretetime dynamics), local-time update flow, and local-time update map. The presence of only bilinear interactions prevent the flow cases to develop any dynamical instability, the system converging always to the thermal equilibrium. The situation is different for the map when unfriendly couplings are involved, where period-two oscillations arise. In the case of the map with local-time updates, oscillations of any period and chaos can arise as a consequence of the reciprocal “tension” accumulated among the communities during their sleeping time interval. The resulting chaos can be of two kinds: true chaos characterized by positive Lyapunov exponent and bifurcation cascades, or marginal chaos characterized by zero Lyapunov exponent and critical continuous regions.

An MILP-MINLP decomposition method for the global optimization of a source based model of the multiperiod blending problem

The multiperiod blending problem involves binary variables and bilinear terms, yielding a nonconvex MINLP. In this work we present two major contributions for the global solution of the problem. The first one is an alternative formulation of the problem. This formulation makes use of redundant constraints that improve the MILP relaxation of the MINLP. The second contribution is an algorithm that decomposes the MINLP model into two levels. The first level, or master problem, is an MILP relaxation of the original MINLP. The second level, or subproblem, is a smaller MINLP in which some of the binary variables of the original problem are fixed. The results show that the new formulation can be solved faster than alternative models, and that the decomposition method can solve the problems faster than state of the art general purpose solvers.

Models and relaxations for the wastewater treatment design problem

The design of industrial wastewater treatment networks is a challenging problem due to the presence of non-convex bilinear terms in the mathematical formulation. Success in finding the global optimal solution is highly dependent on the quality of the relaxation that is used to compute the lower bound for the objective of minimizing the total flowrate going through the treatment units. We derive rigorous relaxations of increasing complexity for the two alternative nonlinear formulations that have been previously proposed. The main focus is on mixed-integer linear programming relaxations from multiparametric disaggregation and on finding out how to select the most convenient set of variables to discretize. Each such relaxation can be used as the basis of a two-stage algorithm for global optimization that iterates over an accuracy parameter that consecutively increases the tightness as well as the size of the underlying problem. Through the solution of sixteen benchmark problems from the literature we show that the computational performance of the algorithm for the best relaxation compares favorably with commercial solvers BARON and GloMIQO.

Bilinear Embedding Label Propagation: Towards Scalable Prediction of Image Labels

Traditional label propagation (LP) is shown to be effective for transductive classification. To enable the standard LP to handle outside images, two inductive methods by label reconstruction or by direct embedding have been presented, of which the latter scheme is relatively more efficient, especially for testing. But almost all inductive LP models use 1D vectors of images as inputs, which may destroy the topology structure of image pixels and usually suffer from high complexity due to the high dimension of 1D vectors in reality. To preserve the topology among pixels and address the scalability issue for the embedding based scheme, we propose a simple yet efficient Bilinear Embedding Label Propagation (BELP) by including a bilinear regularization term in terms of tensor representation to correlate the image labels with their bilinear features. BELP performs label prediction over the 2D matrices rather than 1D vectors, since images are essentially matrices. Finally, labels of new images can be easily obtained by embedding them onto a spanned bilinear subspace solved from a joint framework. Simulations verified the efficiency of our approach.

On a bilinear estimate in weak-Morrey spaces and uniqueness for Navier–Stokes equations

This paper is concerned with the continuity of the bilinear term B associated with the mild formulation of the Navier–Stokes equations. We provide a new proof for the continuity of B in critical weak-Morrey spaces without using auxiliary norms of Besov type neither Kato time-weighted norms. As a byproduct, we reobtain the uniqueness of mild solutions in the class of continuous functions from [0,T) to critical Morrey spaces. Our proof consists in estimates in block spaces (based on Lorentz spaces) that are preduals of Morrey–Lorentz spaces. For that, we need to obtain properties like interpolation of operators, duality, Hölder and Young type inequalities in such block spaces.

Integrated crude selection and refinery optimization under uncertainty

Crude oil selection and procurement is the most important step in the refining process and impacts the profit margin of the refinery significantly. Due to uncertain quality of the crudes, conventional deterministic modeling and optimization methods are not suitable for refinery profitability enhancement. Therefore, a novel optimization scheme for crude oil procurement integrated with refinery operations in the face of uncertainties is presented. The decision process comprises two stages and is solved using a scenario‐based stochastic programming formulation. In Stage I, the optimal crude selections and purchase amounts are determined by maximizing the expected profit across all scenarios. In Stage II, the uncertainties are realized and optimal operations for the refinery are determined according to this realization. The resulting large‐scale mixed‐integer nonlinear programming formulation incorporates integer variables for crude selection and continuous variables for refinery operations, as well as bilinear terms for pooling processes. Nonconvex generalized Benders decomposition is used to solve this problem to obtain an global optimum efficiently. © 2015 American Institute of Chemical Engineers AIChE J, 62: 1038–1053, 2016

Fermionic corrections to fluid dynamics from BTZ black hole

We reconstruct the complete fermionic orbit of the non-extremal BTZ black hole by acting with finite supersymmetry transformations. The solution satisfies the exact supergravity equations of motion to all orders in the fermonic expansion and the final result is given in terms of fermionic bilinears. By fluid/gravity correspondence, we derive linearized Navier-Stokes equations and a set of new differential equations from Rarita-Schwinger equation. We compute the boundary energy-momentum tensor and we interpret the result as a perfect fluid with a modified definition of fluid velocity. Finally, we derive the modified expression for the entropy of the black hole in terms of the fermionic bilinears.

A Bi-Level Branch and Bound Method for Economic Dispatch With Disjoint Prohibited Zones Considering Network Losses

This paper proposes a bi-level branch-and-bound ((B&B) method to solve the economic dispatch problem with prohibited zones and network losses. The approach employs binary variables for each prohibited zone and utilizes the B-coefficient for network losses, which can be transformed into a mixed-integer quadratically constrained quadratic programming (MIQCQP), where linear relaxation technique is applied on each bilinear term. Due to the complexity in solving the MIQCQP problem, this paper proposes a bi-level B&B method to achieve global optimum. A spatial B&B method is utilized in the higher level to solve the quadratically constrained quadratic programming (QCQP) problem, whereas a simple B&B method is employed in the lower level to solve a mixed-integer quadratic programming (MIQP) problem. The bi-level B&B algorithm that combines spatial and simple B&B methods is actually a deterministic optimization method and can produce global optimal solutions. Numerical results on 6-unit, 15-unit, and 40-unit test systems show that the bi-level B&B method can solve the MIQCQP problem with superior solution quality and convergence characteristics.

Transformation-Based Preprocessing for Mixed-Integer Quadratic Programs

This paper presents two preprocessing techniques for mixed-integer quadratic programs with non-convex objective functions, where the continuous part of the Hessian is invertible. The techniques aim at reducing the number of bilinear terms in the objective. Results show that one of the techniques decreases the solution times once the reduction in bilinear terms crosses a threshold.

Effective Solution Approach for Integrated Optimization Models of Refinery Production and Utility System

A typical refinery consists of the production system and the utility system that are routinely optimized in a hierarchical procedure. To meet the demand for higher profit and energy utilization, it is imperative to integrate the two systems for refinery-wide simultaneous optimization, although the integrated model results in a mixed-integer nonlinear programming (MINLP) problem in which the bilinear terms for the correlation of the blending process and gas emission directly result in inconsistency between solution quality and time. The main aim of this work is to propose a solution strategy based on heuristics to decompose the integrated model of the two systems into a mixed-integer linear programming (MILP) model and a nonlinear programming (NLP) model which are then solved iteratively through variables transferring to further reduce the solution time. The solution of the traditional sequential method is incorporated to generate better initial estimates for the decomposed model to gain better solution quality and efficiency. The proposed solution approach is compared with the basic sequential method and the standard MINLP solvers. The results obtained in two scenarios of a case study from a real refinery demonstrate the effectiveness of the proposed decomposition strategy.