Large-scale Polynomial Research Articles

Analog In-memory Circuit Design of Polynomial Multiplication for Lattice Cipher Acceleration Application

As the core operation of lattice cipher, large-scale polynomial multiplication is the biggest computational bottleneck in its realization process. How to quickly calculate polynomial multiplication under resource constraints has become an urgent problem to be solved in the hardware implementation of lattice ciphers. Therefore, an analog in-memory circuit for fast polynomial multiplication calculation is proposed. First, an in-memory computing circuit for Discrete Fourier Transform and Inverse Discrete Fourier Transform based on memristor array is designed. On this basis, a fully analog circuit that can realize polynomial multiplication in one step is designed. Compared with traditional hardware implementation, the in-memory calculation method used in this article decreases the calculation time of polynomial multiplication to the microsecond level, which greatly improves the speed of lattice cipher encryption and decryption. For the specific examples in this article, PSPICE simulation shows that the average accuracy of the calculation result is above 99.90%.

CS-TSSOS: Correlative and Term Sparsity for Large-Scale Polynomial Optimization

This work proposes a new moment-SOS hierarchy, called CS-TSSOS , for solving large-scale sparse polynomial optimization problems. Its novelty is to exploit simultaneously correlative sparsity and term sparsity by combining advantages of two existing frameworks for sparse polynomial optimization. The former is due to Waki et al. [ 40 ] while the latter was initially proposed by Wang et al. [ 42 ] and later exploited in the TSSOS hierarchy [ 46 , 47 ]. In doing so we obtain CS-TSSOS—a two-level hierarchy of semidefinite programming relaxations with (i) the crucial property to involve blocks of SDP matrices and (ii) the guarantee of convergence to the global optimum under certain conditions. We demonstrate its efficiency and scalability on several large-scale instances of the celebrated Max-Cut problem and the important industrial optimal power flow problem, involving up to six thousand variables and tens of thousands of constraints.

Motion generation of a planar 3R serial chain based on conformal geometric algebra with applications to planar linkages

Abstract. A planar three-revolute (3R) serial chain is an important part of many mechanisms. The classical approach in motion generation of a planar 3R serial chain is to construct closed-loop equations based on complex numbers, which yields a large-scale polynomial system. In this study, a new approach of planar 3R serial chain motion generation by establishing the relative kinematics model based on conformal geometric algebra (CGA) is proposed. The simpler design equations are obtained, which can be used to design a planar 3R serial chain that can accurately achieve the N poses. In the numerical examples, the number of different poses is used to verify the correctness and efficiency of the proposed method by using polyhedral homotopy continuation. The results indicate that the design equations obtained via CGA are more concise for improving the solving efficiency compared with the previous method. Finally, a geared five-bar mechanism with a seven-pose motion generation example is considered.

Exploiting term sparsity in noncommutative polynomial optimization

We provide a new hierarchy of semidefinite programming relaxations, called NCTSSOS, to solve large-scale sparse noncommutative polynomial optimization problems. This hierarchy features the exploitation of term sparsity hidden in the input data for eigenvalue and trace optimization problems. NCTSSOS complements the recent work that exploits correlative sparsity for noncommutative optimization problems by Klep et al. (MP, 2021), and is the noncommutative analogue of the TSSOS framework by Wang et al. (SIAMJO 31: 114–141, 2021, SIAMJO 31: 30–58, 2021). We also propose an extension exploiting simultaneously correlative and term sparsity, as done previously in the commutative case (Wang in CS-TSSOS: Correlative and term sparsity for large-scale polynomial optimization, 2020). Under certain conditions, we prove that the optima of the NCTSSOS hierarchy converge to the optimum of the corresponding dense semidefinite programming relaxation. We illustrate the efficiency and scalability of NCTSSOS by solving eigenvalue/trace optimization problems from the literature as well as randomly generated examples involving up to several thousand variables.

A Rational Even-IRA Algorithm for the Solution of $T$-Even Polynomial Eigenvalue Problems

In this work we present a rational Krylov subspace method for solving real large-scale polynomial eigenvalue problems with $T$-even (that is, symmetric/skew-symmetric) structure. Our method is base...

Decentralized Observer-Based Controller Synthesis for a Large-Scale Polynomial T-S Fuzzy System With Nonlinear Interconnection Terms.

This article proposes a novel approach to synthesize a decentralized observer-based controller for a large-scale nonlinear systems in which the interconnection terms are expressed in the nonlinear forms. The large-scale nonlinear system is modeled under the framework of the large-scale polynomial T-S fuzzy systems which can reduce the modeling error and the number of fuzzy rules. It is noted that the interconnection terms are arbitrary nonlinear functions and unnecessary to satisfy the bound constraints, while these constraints are mandatory in the previous studies. Therefore, the proposed method is more relaxed and applicable in practice. A new observer form based on an unknown input method is proposed to simultaneously estimate the unmeasurable states and interconnection terms, which has not been taken into consideration in any previous articles. The information of the unknown states and interconnection is fed back to the controller for stabilizing the system. With the support of the Lyapunov function, the sum-of-square (SOS) technique, the conditions for designing the observer and controller are derived. Finally, the two numerical examples are provided to prove the success and merit of the proposed method.

A polynomial Jacobi–Davidson solver with support for non-monomial bases and deflation

Large-scale polynomial eigenvalue problems can be solved by Krylov methods operating on an equivalent linear eigenproblem (linearization) of size $$d\cdot n$$, where d is the polynomial degree and n is the problem size, or by projection methods that keep the computation in the n-dimensional space. Jacobi–Davidson belongs to the latter class of methods, and, since it is a preconditioned eigensolver, it may be competitive in cases where explicitly computing a matrix factorization is exceedingly expensive. However, a fully fledged implementation of polynomial Jacobi–Davidson has to consider several issues, including deflation to compute more than one eigenpair, use of non-monomial bases for the case of large degree polynomials, and handling of complex eigenvalues when computing in real arithmetic. We discuss these aspects and present computational results of a parallel implementation in the SLEPc library.

Polynomial Controller Synthesis for Uncertain Large-Scale Polynomial T-S Fuzzy Systems.

This paper proposes a novel method to synthesize a controller for stabilizing the nonlinear large-scale system which is represented by a large-scale polynomial Takagi-Sugeno (T-S) fuzzy system. The large-scale system consists of a set of the uncertain polynomial T-S fuzzy system with interconnection terms. Modeling the large-scale nonlinear system under the framework of the polynomial form will decrease both the modeling errors and the number of fuzzy rules with respect to the conventional large-scale T-S fuzzy system. In addition, because of the existence of uncertainties, the synthesizing controller for the large-scale polynomial fuzzy system becomes much more challenging and has not been investigated in the previous studies. In this paper, a controller is synthesized to simultaneously eliminate the impact of the uncertainties and stabilize the system. With the aid of Lyapunov theory, sum-of-square technique, and S-procedure, the conditions for controller synthesis are derived in the main theorems. Finally, two examples are illustrated to show the effectiveness and merit of the proposed method.

Lasserre Hierarchy for Large Scale Polynomial Optimization in Real and Complex Variables

We propose general notions to deal with large scale polynomial optimization problems and demonstrate their efficiency on a key industrial problem of the 21st century, namely the optimal power flow problem. These notions enable us to find global minimizers on instances with up to 4,500 variables and 14,500 constraints. First, we generalize the Lasserre hierarchy from real to complex numbers in order to enhance its tractability when dealing with complex polynomial optimization. Complex numbers are typically used to represent oscillatory phenomena, which are omnipresent in physical systems. Using the notion of hyponormality in operator theory, we provide a finite convergence criterion which generalizes the Curto--Fialkow conditions of the real Lasserre hierarchy. Second, we introduce the multi-ordered Lasserre hierarchy in order to exploit sparsity in polynomial optimization problems (in real or complex variables) while preserving global convergence. It is based on two ideas: (1) to use a different relaxatio...

Sparse-BSOS: a bounded degree SOS hierarchy for large scale polynomial optimization with sparsity

We provide a sparse version of the bounded degree SOS hierarchy BSOS (Lasserre et al. in EURO J Comp Optim:87–117, 2017) for polynomial optimization problems. It permits to treat large scale problems which satisfy a structured sparsity pattern. When the sparsity pattern satisfies the running intersection property this Sparse-BSOS hierarchy of semidefinite programs (with semidefinite constraints of fixed size) converges to the global optimum of the original problem. Moreover, for the class of SOS-convex problems, finite convergence takes place at the first step of the hierarchy, just as in the dense version.

Implicitly Restarted Refined Partially Orthogonal Projection Method with Deflation

AbstractIn this paper we consider the computation of some eigenpairs with smallest eigenvalues in modulus of large-scale polynomial eigenvalue problem. Recently, a partially orthogonal projection method and its refinement scheme were presented for solving the polynomial eigenvalue problem. The methods preserve the structures and properties of the original polynomial eigenvalue problem. Implicitly updating the starting vector and constructing better projection subspace, we develop an implicitly restarted version of the partially orthogonal projection method. Combining the implicit restarting strategy with the refinement scheme, we present an implicitly restarted refined partially orthogonal projection method. In order to avoid the situation that the converged eigenvalues converge repeatedly in the later iterations, we propose a novel explicit non-equivalence low-rank deflation technique. Finally some numerical experiments show that the implicitly restarted refined partially orthogonal projection method with the explicit non-equivalence low-rank deflation technique is efficient and robust.

Random sampling in computational algebra: Helly numbers and violator spaces

This paper transfers a randomized algorithm, originally used in geometric optimization, to computational problems in commutative algebra. We show that Clarkson's sampling algorithm can be applied to two problems in computational algebra: solving large-scale polynomial systems and finding small generating sets of graded ideals. The cornerstone of our work is showing that the theory of violator spaces of Gärtner et al. applies to polynomial ideal problems. To show this, one utilizes a Helly-type result for algebraic varieties. The resulting algorithms have expected runtime linear in the number of input polynomials, making the ideas interesting for handling systems with very large numbers of polynomials, but whose rank in the vector space of polynomials is small (e.g., when the number of variables and degree is constant).

Semidefinite programming relaxation methods for global optimization problems with sparse polynomials and unbounded semialgebraic feasible sets

We propose a hierarchy of semidefinite programming (SDP) relaxations for polynomial optimization with sparse patterns over unbounded feasible sets. The convergence of the proposed SDP hierarchy is established for a class of polynomial optimization problems. This is done by employing known sums-of-squares sparsity techniques of Kojima and Muramatsu Comput Optim Appl 42(1):31---41, (2009) and Lasserre SIAM J Optim 17:822---843, (2006) together with a representation theorem for polynomials over unbounded sets obtained recently in Jeyakumar et al. J Optim Theory Appl 163(3):707---718, (2014). We demonstrate that the proposed sparse SDP hierarchy can solve some classes of large scale polynomial optimization problems with unbounded feasible sets using the polynomial optimization solver SparsePOP developed by Waki et al. ACM Trans Math Softw 35:15 (2008).

On Solving Large-Scale Polynomial Convex Problems by Randomized First-Order Algorithms

One of the most attractive recent approaches to processing well-structured large-scale convex optimization problems is based on smooth convex-concave saddle point reformulation of the problem of interest and solving the resulting problem by a fast first order saddle point method utilizing smoothness of the saddle point cost function. In this paper, we demonstrate that when the saddle point cost function is polynomial, the precise gradients of the cost function required by deterministic first order saddle point algorithms and becoming prohibitively computationally expensive in the extremely large-scale case, can be replaced with incomparably cheaper computationally unbiased random estimates of the gradients. We show that for large-scale problems with favorable geometry, this randomization accelerates, progressively as the sizes of the problem grow, the solution process. This extends significantly previous results on acceleration by randomization, which, to the best of our knowledge, dealt solely with bilinear saddle point problems. We illustrate our theoretical findings by instructive and encouraging numerical experiments.

A Wu‘s Method Based Parallelization Design and Implementation for Algebraic Cryptanalysis

Wus method is one of the effective methods for solving large-scale polynomial equation systems in algebraic cryptanalysis. But it will take a lot of time to solve polynomial equations with a serial algorithm of Wu’s method. In order to eliminate the bottleneck of computing time overheads, we proposes an efficient data parallelization scheme based on Wu‘s method. And a load balance mechanism applied to our scheme, which greatly improves computing-performance of Wu’s method and enhances its ability to meet challenges of algebraic cryptanalysis.

Regularization Methods for SDP Relaxations in Large-Scale Polynomial Optimization

We study how to solve semidefinite programming (SDP) relaxations for large-scale polynomial optimization. When interior-point methods are used, typically only small or moderately large problems could be solved. This paper studies regularization methods for solving polynomial optimization problems. We describe these methods for semidefinite optimization with block structures and then apply them to solve large-scale polynomial optimization problems. The performance is tested on various numerical examples. With regularization methods, significantly bigger problems could be solved on a regular computer, which is almost impossible with interior point methods.

On Handling Large-Scale Polynomial Multiplications in Compute Cloud Environments using Divisible Load Paradigm

Large-scale polynomial product computations often used in aerospace applications such as satellite image processing and sensor networks data processing always pose considerable challenge when processed on networked computing systems. With non-zero communication and computation time delays of the links and processors on a networked infrastructure, the computation becomes all the more challenging. In this research, we attempt to investigate the use of a divisible load paradigm to design efficient strategies to minimize the overall processing time for performing large-scale polynomial product computations in compute cloud environments. We consider a compute cloud system with the resource allocator distributing the entire load to a set of virtual CPU instances (VCI) and the VCIs propagating back the processed results to resource allocator for postprocessing. We consider heterogeneous networks in our analysis and we derive fundamental recursive equations and a closed-form solution for the load fractions to be assigned to each VCI. Our analysis also attempts to eliminate any redundant VCI-link pairs by carefully considering the overheads associated with load distribution and processing. Finally, we quantify the performance of the strategies via rigorous simulation studies.

AN EFFICIENCY STUDY OF POLYNOMIAL EIGENVALUE PROBLEM SOLVERS FOR QUANTUM DOT SIMULATIONS

Nano-scale quantum dot simulations result in large-scale polynomial eigenvalue problems. It remains unclear how these problems can be solved efficiently. We fill this gap in capability partially by proposing a polynomial Jacobi-Davidson method framework, including several varied schemes for solving the associated correction equations. We investigate the performance of the proposed Jacobi-Davidson methods for solving the polynomial eigenvalue problems and several Krylov subspace methods for solving the linear eigenvalue problems with the use of various linear solvers and preconditioning schemes. This study finds the most efficient scheme combinations for different types of target problems.

Restarted generalized Krylov subspace methods for solving large-scale polynomial eigenvalue problems

In this paper, we introduce a generalized Krylov subspace \({\mathcal{G}_{m}(\mathbf{A};\mathbf{u})}\) based on a square matrix sequence {Aj} and a vector sequence {uj}. Next we present a generalized Arnoldi procedure for generating an orthonormal basis of \({\mathcal{G}_{m}(\mathbf{A};\mathbf{u})}\). By applying the projection and the refined technique, we derive a restarted generalized Arnoldi method and a restarted refined generalized Arnoldi method for solving a large-scale polynomial eigenvalue problem (PEP). These two methods are applied to solve the PEP directly. Hence they preserve essential structures and properties of the PEP. Furthermore, restarting reduces the storage requirements. Some theoretical results are presented. Numerical tests report the effectiveness of these methods.

Dynamic Enumeration of All Mixed Cells

The polyhedral homotopy method, which has been known as a powerful numerical method for computing all isolated zeros of a polynomial system, requires all mixed cells of the support of the system to construct a family of homotopy functions. The mixed cells are reformulated in terms of a linear inequality system with an additional combinatorial condition. An enumeration tree is constructed among a family of linear inequality systems induced from it such that every mixed cell corresponds to a unique feasible leaf node, and the depth-first search is applied to the enumeration tree for finding all the feasible leaf nodes. How to construct such an enumeration tree is crucial in computational efficiency. This paper proposes a dynamic construction of an enumeration tree, which branches each parent node into its child nodes so that the number of feasible child nodes is expected to be small; hence we can prune many subtrees which do not contain any mixed cell. Numerical results exhibit that the proposed dynamic construction of an enumeration tree works very efficiently for large scale polynomial systems; for example, it generated all mixed cells of the cyclic-15 problem for the first time in less than 16 hours.