3,029 results on '"Quadratic growth"'
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2. Quadratic Growth and Linear Convergence of a DCA Method for Quartic Minimization over the Sphere.
- Author
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Hu, Shenglong and Yan, Zhifang
- Subjects
- *
EXPONENTS - Abstract
The quartic minimization over the sphere can be reformulated as a nonlinear nonconvex semidefinite program over the spectraplex. In this paper, under mild assumptions, we show that the reformulated nonlinear semidefinite program possesses the quadratic growth property at a rank one critical point which is a local minimizer of the quartic minimization problem. The quadratic growth property further implies the strong metric subregularity of the subdifferential of the objective function of the unconstrained reformulation of the nonlinear semidefinite program, from which we can show that the objective function is a Łojasiewicz function with exponent 1 2 at the corresponding critical point. With these results, we can establish the linear convergence of an efficient DCA method proposed for solving the nonlinear semidefinite program. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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3. Sharper Analysis for Minibatch Stochastic Proximal Point Methods: Stability, Smoothness, and Deviation.
- Author
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Xiao-Tong Yuan and Ping Li
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- *
STOCHASTIC analysis , *STABILITY theory , *REGRESSION analysis , *PARAMETER estimation , *OVERHEAD costs , *LOGISTIC regression analysis - Abstract
The stochastic proximal point (SPP) methods have gained recent attention for stochastic optimization, with strong convergence guarantees and superior robustness to the classic stochastic gradient descent (SGD) methods showcased at little to no cost of computational overhead added. In this article, we study a minibatch variant of SPP, namely M-SPP, for solving convex composite risk minimization problems. The core contribution is a set of novel excess risk bounds of M-SPP derived through the lens of algorithmic stability theory. Particularly under smoothness and quadratic growth conditions, we show that M-SPP with minibatch-size n and iteration count T enjoys an in-expectation fast rate of convergence consisting of an ... bias decaying term and an ... variance decaying term. In the small-n-large-T setting, this result substantially improves the best known results of SPP-type approaches by revealing the impact of noise level of model on convergence rate. In the complementary small-T-large-n regime, we propose a two-phase extension of M-SPP to achieve comparable convergence rates. Additionally, we establish a deviation bound on the parameter estimation error of a sampling-without-replacement variant of M-SPP, which holds with high probability over the randomness of data while in expectation over the randomness of algorithm. Numerical evidences are provided to support our theoretical predictions when substantialized to Lasso and logistic regression models. [ABSTRACT FROM AUTHOR]
- Published
- 2023
4. Backward doubly stochastic differential equations and SPDEs with quadratic growth.
- Author
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Hu, Ying, Wen, Jiaqiang, and Xiong, Jie
- Subjects
- *
QUADRATIC equations , *SOBOLEV spaces , *STOCHASTIC differential equations - Abstract
This paper shows the nonlinear stochastic Feynman–Kac formula holds under quadratic growth. For this, we initiate the study of backward doubly stochastic differential equations (BDSDEs, for short) with quadratic growth. The existence, uniqueness, and comparison theorem for one-dimensional BDSDEs are proved when the generator f (t , Y , Z) grows in Z quadratically and the terminal value is bounded, by introducing innovative approaches. Furthermore, in this framework, we utilize BDSDEs to provide a probabilistic representation of solutions to semilinear stochastic partial differential equations (SPDEs, for short) in Sobolev spaces, and use it to prove the existence and uniqueness of such SPDEs, thereby extending the nonlinear stochastic Feynman–Kac formula for linear growth introduced by Pardoux and Peng (1994). [ABSTRACT FROM AUTHOR]
- Published
- 2024
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5. Quadratic Growth and Strong Metric Subregularity of the Subdifferential for a Class of Non-prox-regular Functions.
- Author
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Chieu, Nguyen Huy, Trang, Nguyen Thi Quynh, and Tuan, Ha Anh
- Subjects
- *
DIFFERENTIABLE functions , *SENSES - Abstract
This paper mainly studies the quadratic growth and the strong metric subregularity of the subdifferential of a function that can be represented as the sum of a function twice differentiable in the extended sense and a subdifferentially continuous, prox-regular, twice epi-differentiable function. For such a function, which is not necessarily prox-regular, it is shown that the quadratic growth, the strong metric subregularity of the subdifferential at a local minimizer, and the positive definiteness of the subgradient graphical derivative at a stationary point are equivalent. In addition, other characterizations of the quadratic growth and the strong metric subregularity of the subdifferential are also given. Besides, properties of functions twice differentiable in the extended sense are examined. [ABSTRACT FROM AUTHOR]
- Published
- 2022
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6. Parabolic Equations with Quadratic Growth in
- Author
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Bensoussan, Alain, Frehse, Jens, Peng, Shige, Yam, Sheung Chi Phillip, Oñate, Eugenio, Series Editor, Chetverushkin, B. N., editor, Fitzgibbon, W., editor, Kuznetsov, Y.A., editor, Neittaanmäki, P., editor, Periaux, J., editor, and Pironneau, O., editor
- Published
- 2019
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7. Representation of solutions to quadratic 2BSDEs with unbounded terminal values.
- Author
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Kim, Kon-Gun, Kim, Mun-Chol, and Hwang, Ho-Jin
- Subjects
- *
STOCHASTIC orders , *QUADRATIC forms , *STOCHASTIC differential equations , *MARTINGALES (Mathematics) - Abstract
Second order backward stochastic differential equations (2BSDEs, for short) are one of useful tools in solving stochastic control problems with model uncertainty. In this paper, we prove a representation formula for quadratic 2BSDEs with an unbounded terminal value under a convex assumption on the generator. Because of the unboundedness of the terminal value, we are unable to use some fine properties of BMO martingales, which are often employed in the literature to deal with bounded solutions to quadratic backward stochastic differential equations. Instead, we utilize the θ -technique. We also prove an existence result under an additional assumption that the terminal value is of uniformly continuous. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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8. New nonasymptotic convergence rates of stochastic proximal point algorithm for stochastic convex optimization.
- Author
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Pătraşcu, Andrei
- Subjects
- *
STOCHASTIC convergence , *CONVEX functions , *MACHINE learning , *STOCHASTIC programming , *CONVEX programming - Abstract
Large sectors of the recent optimization literature focused in the last decade on the development of optimal stochastic first-order schemes for constrained convex models under progressively relaxed assumptions. Stochastic proximal point is an iterative scheme born from the adaptation of proximal point algorithm to noisy stochastic optimization, with a resulting iteration related to stochastic alternating projections. Inspired by the scalability of alternating projection methods, we start from the (linear) regularity assumption, typically used in convex feasiblity problems to guarantee the linear convergence of stochastic alternating projection methods, and analyze a general weak linear regularity condition which facilitates convergence rate boosts in stochastic proximal point schemes. Our applications include many non-strongly convex functions classes often used in machine learning and statistics. Moreover, under weak linear regularity assumption we guarantee O 1 k convergence rate for SPP, in terms of the distance to the optimal set, using only projections onto a simple component set. Linear convergence is obtained for interpolation setting, when the optimal set of the expected cost is included into the optimal sets of each functional component. [ABSTRACT FROM AUTHOR]
- Published
- 2021
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9. Existence and regularity results for nonlinear parabolic equations with quadratic growth with respect to the gradient.
- Author
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Marah, Amine, Redwane, H., and Zaki, K.
- Abstract
In this paper we study the existence and regularity of solutions of nonlinear parabolic equations of the type ∂ u ∂ t - div ((a (x , t) + | u | q ) ∇ u) + b (x , t) u | u | p - 1 | ∇ u | 2 = f in Q , u (t = 0) = 0 in Ω , u = 0 on ∂ Ω × (0 , T) , where Q = Ω × (0 , T) , Ω is a bounded open subset of R N (N > 2) , a(x, t), b(x, t) are measurable positive functions, p , q > 0 and f belongs to L m (Q) for some m ≥ 1 . [ABSTRACT FROM AUTHOR]
- Published
- 2021
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10. BSDEs with Quadratic Growth in Z
- Author
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Zhang, Jianfeng, Glynn, Peter W., Editor-in-chief, Kyprianou, Andreas E., Editor-in-chief, Le Jan, Yves, Editor-in-chief, Asmussen, Søren, Series editor, Hairer, Martin, Series editor, Jagers, Peter, Series editor, Karatzas, Ioannis, Series editor, Kelly, Frank P., Series editor, Øksendal, Bernt, Series editor, Papanicolaou, George, Series editor, Pardoux, Etienne, Series editor, Perkins, Edwin, Series editor, Soner, Halil Mete, Series editor, and Zhang, Jianfeng
- Published
- 2017
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11. QUADRATIC GROWTH AND STRONG METRIC SUBREGULARITY OF THE SUBDIFFERENTIAL VIA SUBGRADIENT GRAPHICAL DERIVATIVE.
- Author
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NGUYEN HUY CHIEU, LE VAN HIEN, TRAN T. A. NGHIA, and HA ANH TUAN
- Subjects
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SUBDIFFERENTIALS , *SET functions , *FINITE, The - Abstract
This paper mainly studies the relationship between quadratic growth and strong metric subregularity of the subdifferential in finite dimensional settings by using the subgradient graphical derivative. We prove that the positive definiteness of the subgradient graphical derivative of an extended-real-valued lower semicontinuous proper function at a proximal stationary point is sufficient for the point to be a local minimizer at which the subdifferential is strongly subregular for 0. The latter was known to imply the quadratic growth. When the function is either subdifferentially continuous, prox-regular, twice epidifferentiable, or variationally convex, we show that the quadratic growth, the strong metric subregularity of the subdifferential at a local minimizer, and the positive definiteness of the subgradient graphical derivative at a stationary point are equivalent. For C²-cone reducible constrained programs satisfying the metric subregularity constraint qualification, we obtain the same results for the sum of the objective function and the indicator function of the feasible set. [ABSTRACT FROM AUTHOR]
- Published
- 2021
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12. Quadratic growth during the COVID-19 pandemic : merging hotspots and reinfections
- Abstract
The existence of an exponential growth phase during early stages of a pandemic is often taken for granted. However, for the 2019 novel coronavirus epidemic, the early exponential phase lasted only for about six days, while the quadratic growth prevailed for forty days until it spread to other countries and continued, again quadratically, but with a shorter time constant. Here we show that this rapid phase is followed by a subsequent slow-down where the coefficient is reduced to almost the original value at the outbreak. This can be explained by the merging of previously disconnected sites that occurred after the disease jumped (nonlocally) to a relatively small number of separated sites. Subsequent variations in the slope with continued growth can qualitatively be explained as a result of reinfections and variations in their rate. We demonstrate that the observed behavior can be described by a standard epidemiological model with spatial extent and reinfections included. Time-dependent changes in the spatial diffusion coefficient can also model corresponding variations in the slope., QC 20230316
- Published
- 2023
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13. Quadratic growth during the COVID-19 pandemic : merging hotspots and reinfections
- Abstract
The existence of an exponential growth phase during early stages of a pandemic is often taken for granted. However, for the 2019 novel coronavirus epidemic, the early exponential phase lasted only for about six days, while the quadratic growth prevailed for forty days until it spread to other countries and continued, again quadratically, but with a shorter time constant. Here we show that this rapid phase is followed by a subsequent slow-down where the coefficient is reduced to almost the original value at the outbreak. This can be explained by the merging of previously disconnected sites that occurred after the disease jumped (nonlocally) to a relatively small number of separated sites. Subsequent variations in the slope with continued growth can qualitatively be explained as a result of reinfections and variations in their rate. We demonstrate that the observed behavior can be described by a standard epidemiological model with spatial extent and reinfections included. Time-dependent changes in the spatial diffusion coefficient can also model corresponding variations in the slope.
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- 2023
- Full Text
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14. On the Gradient Projection Method for Weakly Convex Functions on a Proximally Smooth Set.
- Author
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Balashov, M. V.
- Subjects
- *
SMOOTHNESS of functions , *NONSMOOTH optimization , *SET functions , *CONVEX functions - Abstract
Let a weakly convex function (in the general case, nonconvex and nonsmooth) satisfy the quadratic growth condition. It is proved that the gradient projection method for minimizing such a function on a set converges with linear rate on a proximally smooth (nonconvex) set of special form (for example, on a smooth manifold), provided that the constant of weak convexity of the function is less than the constant in the quadratic growth condition and the constant of proximal smoothness for the set is sufficiently large. The connection between the quadratic growth condition on the function and other conditions is discussed. [ABSTRACT FROM AUTHOR]
- Published
- 2020
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15. Extensions of Gronwall's inequality with quadratic growth terms and applications
- Author
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Jeff Webb
- Subjects
gronwall inequality ,quadratic growth ,second order equation ,Mathematics ,QA1-939 - Abstract
We obtain some new Gronwall type inequalities where, instead of linear growth assumptions, we allow quadratic (or more) growth provided some additional conditions are satisfied. Applications are made to both local and nonlocal boundary value problems for some second order ordinary differential equations which have quadratic growth in the derivative terms.
- Published
- 2018
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16. Invariant representation for generators of general time interval quadratic BSDEs under stochastic growth conditions.
- Author
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Zhou, Guangshuo, Du, Fengjiao, and Fan, Shengjun
- Abstract
This paper is devoted to proving a general invariant representation theorem for generators of general time interval backward stochastic differential equations, where the generator g has a quadratic growth in the unknown variable z and satisfies some stochastic growth conditions in the unknown variable y. This unifies and strengthens some known results. A natural and innovative idea is used to prove the representation theorem. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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17. Quadratic reflected BSDEs and related obstacle problems for PDEs.
- Author
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Huang, Zongyuan, Wang, Haiyang, Wu, Zhen, and Yu, Zhiyong
- Subjects
- *
STOCHASTIC differential equations , *PARTIAL differential equations , *QUADRATIC equations - Abstract
In this paper, we study a kind of reflected backward stochastic differential equations (BSDEs) whose generators are of quadratic growth in z and linear growth in y. We first give an estimate of solutions to such reflected BSDEs. Then under the condition that the generators are convex with respect to z, we can obtain a comparison theorem, which implies the uniqueness of solutions for this kind of reflected BSDEs. Besides, the assumption of convexity also leads to a stability property in the spirit of above estimate. We further establish the nonlinear Feynman-Kac formula of the related obstacle problems for partial differential equations (PDEs) in our framework. At last, a numerical example is given to illustrate the applications of our theoretical results, as well as its connection with an optimal stopping time problem. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
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18. Anticipated backward SDEs with jumps and quadratic-exponential growth drivers.
- Author
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Fujii, Masaaki and Takahashi, Akihiko
- Subjects
- *
STOCHASTIC differential equations , *EXPONENTIAL functions , *DIFFUSION coefficients , *JUMPING - Abstract
In this paper, we study a class of Anticipated Backward Stochastic Differential Equations (ABSDE) with jumps. The solution of the ABSDE is a triple (Y , Z , ψ) where Y is a semimartingale, and (Z , ψ) are the diffusion and jump coefficients. We allow the driver of the ABSDE to have linear growth on the uniform norm of Y 's future paths, as well as quadratic and exponential growth on the spot values of (Z , ψ) , respectively. The existence of the unique solution is proved for Markovian and non-Markovian settings with different structural assumptions on the driver. In the former case, some regularities on (Z , ψ) with respect to the forward process are also obtained. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
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19. A priori bounds and multiplicity for fully nonlinear equations with quadratic growth in the gradient.
- Author
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Nornberg, Gabrielle and Sirakov, Boyan
- Subjects
- *
NONLINEAR equations , *ELLIPTIC equations , *MATHEMATICAL bounds , *MULTIPLICITY (Mathematics) , *BOUNDARY value problems - Abstract
Abstract We consider fully nonlinear uniformly elliptic equations with quadratic growth in the gradient, such as − F (x , u , D u , D 2 u) = λ c (x) u + 〈 M (x) D u , D u 〉 + h (x) in a bounded domain with a Dirichlet boundary condition; here λ ∈ R , c , h ∈ L p (Ω) , p > n ≥ 1 , c ≩ 0 and the matrix M satisfies 0 < μ 1 I ≤ M ≤ μ 2 I. Recently this problem was studied in the "coercive" case λ c ≤ 0 , where uniqueness of solutions can be expected; and it was conjectured that the solution set is more complex for noncoercive equations. This conjecture was verified in 2015 by Arcoya, de Coster, Jeanjean and Tanaka for equations in divergence form, by exploiting the integral formulation of the problem. Here we show that similar phenomena occur for general, even fully nonlinear, equations in nondivergence form. We use different techniques based on the maximum principle. We develop a new method to obtain the crucial uniform a priori bounds, which permit to us to use degree theory. This method is based on basic regularity estimates such as half-Harnack inequalities, and on a Vázquez type strong maximum principle for our kind of equations. [ABSTRACT FROM AUTHOR]
- Published
- 2019
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- View/download PDF
20. Exact LMI Conditions for Stability and $\mathcal {L}_2$ Gain Analysis of 2-D Mixed Continuous–Discrete Time Systems via Quadratically Frequency-Dependent Lyapunov Functions
- Author
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Graziano Chesi
- Subjects
Lyapunov function ,Quadratic growth ,Generalization ,Linear matrix inequality ,Stability (probability) ,Upper and lower bounds ,Computer Science Applications ,symbols.namesake ,Discrete time and continuous time ,Control and Systems Engineering ,Structural stability ,symbols ,Applied mathematics ,Electrical and Electronic Engineering ,Mathematics - Abstract
This paper addresses the problems of establishing structural stability and the L2 gain of 2D mixed continuous-discrete-time systems. The first contribution is to show that Lyapunov functions quadratically dependent on the frequency are exact for establishing structural stability. This is particularly important since the existing works that exploit Lyapunov functions provide a much larger upper bound on the dependence on the frequency or other parameters. The second contribution is to propose a novel linear matrix inequality (LMI) necessary and sufficient condition for establishing the existence of such Lyapunov functions. It is shown, analytically and through several examples, for both best and worst cases, that the numerical complexity of this novel condition is much smaller than that of the existing methods. The third contribution is to show that the proposed methodology can be used to establish upper bounds on the L2 gain, in particular, deriving a novel necessary and sufficient LMI condition based on Lyapunov functions quadratically dependent on the frequency. Lastly, the paper presents the generalization of the proposed methodology to non-mixed 2D systems.
- Published
- 2022
21. Quadratic backward stochastic differential equations driven by [formula omitted]-Brownian motion: Discrete solutions and approximation.
- Author
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Hu, Ying, Lin, Yiqing, and Soumana Hima, Abdoulaye
- Subjects
- *
STOCHASTIC differential equations , *BROWNIAN motion , *APPROXIMATION theory , *BANACH spaces , *COEFFICIENTS (Statistics) - Abstract
Abstract In this paper, we consider backward stochastic differential equations driven by G -Brownian motion (GBSDEs) under quadratic assumptions on coefficients. We prove the existence and uniqueness of solution for such equations. On the one hand, a priori estimates are obtained by applying the Girsanov type theorem in the G -framework, from which we deduce the uniqueness. On the other hand, to prove the existence of solutions, we first construct solutions for discrete GBSDEs by solving corresponding fully nonlinear PDEs, and then approximate solutions for general quadratic GBSDEs in Banach spaces. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
22. Bounded solutions for general time interval BSDEs with quadratic growth coefficients and stochastic conditions.
- Author
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Luo, Huan-Huan and Fan, Sheng-Jun
- Subjects
- *
STOCHASTIC differential equations , *STOCHASTIC analysis , *UNIQUENESS (Mathematics) , *COEFFICIENTS (Statistics) , *QUADRATIC fields - Abstract
This paper deals with bounded solutions for general time interval one-dimensional backward stochastic differential equations (BSDEs for short) with quadratic growth coefficients and stochastic conditions. Several general results of existence, uniqueness, stability and comparison for the bounded solutions are put forward and established, which improve considerably some existing works, even though for the case of finite time interval. Some new ideas are also developed to establish these results. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
23. An Explicit Rate-Optimal Streaming Code for Channels With Burst and Arbitrary Erasures
- Author
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Ashish Khisti, Elad Domanovitz, and Silas L. Fong
- Subjects
FOS: Computer and information sciences ,Quadratic growth ,Sequence ,Computer science ,Information Theory (cs.IT) ,Computer Science - Information Theory ,Packet erasure channel ,020206 networking & telecommunications ,02 engineering and technology ,Data_CODINGANDINFORMATIONTHEORY ,Library and Information Sciences ,Topology ,Computer Science Applications ,Transmission (telecommunications) ,Sliding window protocol ,0202 electrical engineering, electronic engineering, information engineering ,Field size ,Code (cryptography) ,Decoding methods ,Computer Science::Information Theory ,Information Systems ,Communication channel - Abstract
This paper considers the transmission of an infinite sequence of messages (a streaming source) over a packet erasure channel, where every source message must be recovered perfectly at the destination subject to a fixed decoding delay. While the capacity of a channel that introduces only bursts of erasures is well known, only recently, the capacity of a channel with either one burst of erasures or multiple arbitrary erasures in any fixed-sized sliding window has been established. However, the codes shown to achieve this capacity are either non-explicit constructions (proven to exist) or explicit constructions that require large field size that scales exponentially with the delay. This work describes an explicit rate-optimal construction for admissible channel and delay parameters over a field size that scales only quadratically with the delay., arXiv admin note: text overlap with arXiv:1903.07434
- Published
- 2022
24. Semi active damping force estimation using LPV−H∞ estimators with different sensing configurations
- Author
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Juan C. Tudon-Martinez, Olivier Sename, Luis Amezquita-Brooks, and Diana Hernandez-Alcantara
- Subjects
Quadratic growth ,Computer Networks and Communications ,Computer science ,Frequency band ,Applied Mathematics ,Estimator ,Measure (mathematics) ,Damper ,Variable (computer science) ,Hysteresis ,Control and Systems Engineering ,Control theory ,Signal Processing ,Instrumentation (computer programming) - Abstract
Semi-active suspension systems have become a widespread tool to improve the handling and comfort of vehicles. These systems require adjustable dampers as well as supplementary sensing elements. In addition to displacements and accelerations, some of the best performing approaches require knowledge of the semi active damper force. Since this variable can be difficult and expensive to measure, several estimation methods have been proposed. In this article, two Linear-Parameter-Varying H ∞ (LPV- H ∞ ) filters are developed to estimate the Semi-Active (SA) damper force, considering two different combinations of sensing elements: the first configuration is more expensive, but potentially more accurate and reliable; whereas the second configuration is cheaper and arguably less reliable. Thanks to the use of LPV- H ∞ theory, both filters are designed to account for the main nonlinear phenomena of SA dampers (i.e. saturation, hysteresis, etc.), as well as being quadratically stable, robust to the road disturbances and optimized to reduce the estimation error in a specified frequency band. Simulations and experimental data are used to assess the proposed estimators as well as a typical inverse-dynamics estimation approach. The results show that while both of the proposed estimators yield a good degree of accuracy, there are indeed fundamental differences depending on the available sensing elements; a conclusion which could be crucial to appropriately define the instrumentation of semi-active suspension systems.
- Published
- 2022
25. Existence of bounded solutions for quasilinear parabolic systems with quadratic growth
- Author
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Rezak Souilah
- Subjects
Quasilinear parabolic systems ,quadratic growth ,upper and lower solutions ,bounded solutions ,Mathematics ,QA1-939 - Abstract
Assuming the existence of an upper and a lower solution, we prove the existence of at least one bounded solution of a quasilinear parabolic systems, with nonlinear second member having a quadratic growth with respect to the gradient of the solution.
- Published
- 2016
26. Characterization of quadratic growth of extended-real-valued functions
- Author
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Jin jiang Wang and Wen Song
- Subjects
quadratic growth ,Mordukhovich (limiting) subdifferential ,strong metric regularity ,strong metric subregularity ,prox-regular ,Mathematics ,QA1-939 - Abstract
Abstract This paper shows that the sharpest possible bound in the second-order growth condition of a proper lower semicontinuous function can be attained under some assumptions. We also establish a relationship among strong metric subregularity, quadratic growth, the positive-definiteness property of the second-order subdifferential/generalized Hessian, the strong metric regularity, and tilt stability in a finite-dimensional setting.
- Published
- 2016
- Full Text
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27. Constrained Online Convex Optimization With Feedback Delays
- Author
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Junshan Zhang, Xuanyu Cao, and H. Vincent Poor
- Subjects
Quadratic growth ,Mathematical optimization ,Sublinear function ,Control and Systems Engineering ,Complete information ,Computer science ,Saddle point ,Convex optimization ,A priori and a posteriori ,Regret ,Electrical and Electronic Engineering ,Convex function ,Computer Science Applications - Abstract
In this article, we study constrained online convex optimization (OCO) in the presence of feedback delays, where a decision maker chooses sequential actions without knowing the loss functions and constraint functions a priori . The loss/constraint functions vary with time and their feedback information is revealed to the decision maker with delays, which arise in many applications. We first consider the scenario of delayed function feedback, in which the complete information of the loss/constraint functions is revealed to the decision maker with delays. We develop a modified online saddle point algorithm suitable for constrained OCO with feedback delays. Sublinear regret and sublinear constraint violation bounds are established for the algorithm in terms of the delays. In practice, the complete information (functional forms) of the loss/constraint functions may not be revealed to the decision maker. Thus, we further examine the scenario of delayed bandit feedback, where only the values of the loss/constraint functions at two random points close to the chosen action are revealed to the decision maker with delays. A delayed version of the bandit online saddle point algorithm is proposed by utilizing stochastic gradient estimates of the loss/constraint functions based on delayed bandit feedback. We also establish sublinear regret and sublinear constraint violation bounds for this bandit optimization algorithm in terms of the delays. Finally, numerical results for online quadratically constrained quadratic programs are presented to corroborate the efficacy of the proposed algorithms.
- Published
- 2021
28. Basic Numerical Iterative Methods
- Author
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Bing Hao Lin, Victor Andrean, Ryan Kuo-Lung Lian, and Ramadhani Kurniawan Subroto
- Subjects
Quadratic growth ,Nonlinear system ,Quadratic equation ,Computer science ,Iterative method ,Numerical analysis ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,Convergence (routing) ,MathematicsofComputing_NUMERICALANALYSIS ,Particle swarm optimization ,Applied mathematics ,Square (algebra) - Abstract
A numerical method is a mathematical tool designed to solve numerical problems. The implementation of a numerical method in a programming language is called a numerical algorithm. This chapter provides a basic review of some of the commonly used iterative numerical algorithms for solving nonlinear problems. It briefly goes through the algorithms of Gauss–Seidel, predictor‐corrector, Newton’s method, and the particle swarm optimization algorithm. The chapter also summarizes the approximate errors for Gauss–Seidel and Newton’s methods. It clearly shows that the error of Newton’s method decreases quadratically (square of the error) whereas that of the Gauss–Seidel method decreases linearly. Newton’s method is a very powerful method as the convergence of the solution is quadratic, which means that the difference between the root and the approximation is squared at each iteration step.
- Published
- 2021
29. Rouben Ranking Function and parametric approach to quadratically constrained multiobjective quadratic fractional programming with trapezoidal fuzzy number coefficients
- Author
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Vandana Goyal, Deepak Gupta, and Namrata Rani
- Subjects
Quadratic growth ,Quadratic equation ,Fractional programming ,Ranking ,Computer science ,Strategy and Management ,Applied mathematics ,Fuzzy number ,Function (mathematics) ,Safety, Risk, Reliability and Quality ,Parametric statistics - Published
- 2021
30. Robust numerical method for singularly perturbed semilinear parabolic differential difference equations
- Author
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Masho Jima Kabeto and Gemechis File Duressa
- Subjects
Quadratic growth ,Numerical Analysis ,Partial differential equation ,General Computer Science ,Discretization ,Differential equation ,Applied Mathematics ,Numerical analysis ,Finite difference ,Domain (mathematical analysis) ,Theoretical Computer Science ,Algebraic equation ,Modeling and Simulation ,Applied mathematics ,Mathematics - Abstract
This paper deals with the robust numerical method for solving the singularly perturbed semilinear partial differential equation with the spatial delay. The quadratically convergent quasilinearization technique is used to linearize the semilinear term. It is formulated by discretization of the solution domain and then replacing the differential equation by finite difference approximation that in turn gives the system of algebraic equations. The method is shown to be first-order convergent. It is observed that the convergence is independent of the perturbation parameter. Numerical illustrations are investigated on model examples to support the theoretical results and the effectiveness of the method.
- Published
- 2021
31. Backward SDEs
- Author
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Cvitanić, Jakša, Zhang, Jianfeng, Avellaneda, M., editor, Barone-Adesi, G., editor, Broadie, M., editor, Davis, M.H.A., editor, Derman, E., editor, Klüppelberg, C., editor, Schachermayer, W., editor, Cvitanić, Jakša, and Zhang, Jianfeng
- Published
- 2013
- Full Text
- View/download PDF
32. A Scalable EM Algorithm for Hawkes Processes
- Author
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Halpin, Peter F., Millsap, Roger E., editor, van der Ark, L. Andries, editor, Bolt, Daniel M., editor, and Woods, Carol M., editor
- Published
- 2013
- Full Text
- View/download PDF
33. Quadratic Backward SDEs
- Author
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Touzi, Nizar and Touzi, Nizar
- Published
- 2013
- Full Text
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34. The role of conservation laws in the analysis of conformally invariant problems
- Author
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Rivière, Tristan and Mingione, Giuseppe, editor
- Published
- 2012
- Full Text
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35. Observer-Based $$H_\infty $$ Control for One-Sided Lipschitz Nonlinear Systems with Uncertain Input Matrix
- Author
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Seyed Jalil Sadati, Majid Shahbazzadeh, and Homa Salehifar
- Subjects
Quadratic growth ,Matrix (mathematics) ,Observer (quantum physics) ,Relation (database) ,Applied Mathematics ,Signal Processing ,Order (ring theory) ,Bilinear interpolation ,Applied mathematics ,Lipschitz continuity ,Parametric statistics ,Mathematics - Abstract
This paper investigates the problem of observer-based $$H_\infty $$ control for one-sided Lipschitz nonlinear systems subject to parametric uncertainties and external disturbances. In order to relax some conservatisms and limitations of the traditional Lipschitz condition, the one-sided Lipschitz and quadratically inner-bounded conditions are used. On the contrary to the methods proposed in the literature, our method allows for uncertainty in the input matrix B, as well as the dynamic and output matrices A and C. To derive design conditions in terms of LMIs, the well-known Young’s relation is employed for handling the bilinear terms naturally arising in observer-based controller design. Finally, two examples are presented to demonstrate the validity of the theoretical results.
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- 2021
36. On the local stability of semidefinite relaxations
- Author
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Sameer Agarwal, Rekha R. Thomas, Diego Cifuentes, and Pablo A. Parrilo
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Quadratic growth ,Semidefinite programming ,Optimization problem ,90C22 (Primary), 90C31 (Secondary) ,General Mathematics ,Triangulation (social science) ,Mathematics - Algebraic Geometry ,Quadratic equation ,Optimization and Control (math.OC) ,FOS: Mathematics ,Applied mathematics ,Relaxation (approximation) ,Parametric family ,Algebraic Geometry (math.AG) ,Mathematics - Optimization and Control ,Rotation (mathematics) ,Software ,Mathematics - Abstract
We consider a parametric family of quadratically constrained quadratic programs (QCQP) and their associated semidefinite programming (SDP) relaxations. Given a nominal value of the parameter at which the SDP relaxation is exact, we study conditions (and quantitative bounds) under which the relaxation will continue to be exact as the parameter moves in a neighborhood around the nominal value. Our framework captures a wide array of statistical estimation problems including tensor principal component analysis, rotation synchronization, orthogonal Procrustes, camera triangulation and resectioning, essential matrix estimation, system identification, and approximate GCD. Our results can also be used to analyze the stability of SOS relaxations of general polynomial optimization problems., 23 pages, 3 figures
- Published
- 2021
37. A smoothing Levenberg-Marquardt method for the complementarity problem over symmetric cone
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Sanyang Liu and Xiangjing Liu
- Subjects
Levenberg–Marquardt algorithm ,Quadratic growth ,Nonlinear system ,symbols.namesake ,Complementarity theory ,Jacobian matrix and determinant ,symbols ,Applied mathematics ,Function (mathematics) ,Lipschitz continuity ,Smoothing ,Mathematics - Abstract
In this paper, we propose a smoothing Levenberg-Marquardt method for the symmetric cone complementarity problem. Based on a smoothing function, we turn this problem into a system of nonlinear equations and then solve the equations by the method proposed. Under the condition of Lipschitz continuity of the Jacobian matrix and local error bound, the new method is proved to be globally convergent and locally superlinearly/quadratically convergent. Numerical experiments are also employed to show that the method is stable and efficient.
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- 2021
38. Asymptotic fiber orientation states of the quadratically closed Folgar–Tucker equation and a subsequent closure improvement
- Author
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Tobias Karl, Thomas Böhlke, Davide Gatti, and Bettina Frohnapfel
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Quadratic growth ,Velocity gradient ,Fiber (mathematics) ,Mechanical Engineering ,Mathematical analysis ,Closure (topology) ,Orientation (graph theory) ,Condensed Matter Physics ,Quadratic equation ,Mechanics of Materials ,General Materials Science ,Tensor ,ddc:620 ,Anisotropy ,Engineering & allied operations ,Mathematics - Abstract
Anisotropic fiber-reinforced composites are used in lightweight construction, which is of great industrial relevance. During mold filling of fiber suspensions, the microstructural evolution of the local fiber arrangement and orientation distribution is determined by the local velocity gradient. Based on the Folgar–Tucker equation, which describes the evolution of the second-order fiber orientation tensor in terms of the velocity gradient, the present study addresses selected states of deformation rates that can locally occur in complex flow fields. For such homogeneous flows, exact solutions for the asymptotic fiber orientation states are derived and discussed based on the quadratic closure. In contrast to the existing literature, the derived exact solutions take into account the fiber-fiber interaction. The analysis of the asymptotic solutions relying upon the common quadratic closure shows disadvantages with respect to the predicted material symmetry, namely, the anisotropy is overestimated for strong fiber-fiber interaction. This motivates us to suggest a novel normalized fully symmetric quadratic closure. Two versions of this new closure are derived regarding the prediction of anisotropic properties and the fiber orientation evolution. The fiber orientation states determined with the new closure approach show an improved prediction of anisotropy in both effective viscous and elastic composite behaviors. In addition, the symmetrized quadratic closure has a simple structure that reduces the effort in numerical implementation compared to more elaborated closure schemes.
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- 2021
39. Ellipsoidal one-class constraint acquisition for quadratically constrained programming
- Author
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Tomasz P. Pawlak and Bartosz Litwiniuk
- Subjects
Quadratic growth ,050210 logistics & transportation ,021103 operations research ,Information Systems and Management ,General Computer Science ,Computer science ,05 social sciences ,0211 other engineering and technologies ,02 engineering and technology ,Management Science and Operations Research ,Ellipsoid ,Industrial and Manufacturing Engineering ,Modeling and Simulation ,0502 economics and business ,Principal component analysis ,Quadratic programming ,Algorithm - Abstract
We propose Ellipsoidal One-Class Constraint Acquisition (EOCCA), a fast and scalable algorithm for the acquisition of constraints for Mixed-Integer Quadratically Constrained Programming (MIQCP) models from data. EOCCA acquires a well-formed MIQCP model using solely the examples of the feasible solutions to this model. It combines x-means partitioning, standardization, and principal components analysis to preprocess the training set and then wraps the preprocessed data into several hyper-ellipsoids expressed using MIQCP constraints. These MIQCP constraints are projected back to the space of the original training set, and their further use does not require data preprocessing. Experimental evaluation shows that EOCCA scores better than a state-of-the-art algorithm in terms of fidelity of the acquired constraints to ground-truth constraints and achieves this in few orders of magnitude shorter time. We demonstrate the practical use case of EOCCA in a fully automated workflow of modeling and optimization of a rice farm using real-world data.
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- 2021
40. Maximizing perturbation radii for robust convex quadratically constrained quadratic programs
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Pengfei Yu, Ruotian Gao, and Wenxun Xing
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Quadratic growth ,050210 logistics & transportation ,021103 operations research ,Information Systems and Management ,General Computer Science ,05 social sciences ,MathematicsofComputing_NUMERICALANALYSIS ,0211 other engineering and technologies ,02 engineering and technology ,Management Science and Operations Research ,Industrial and Manufacturing Engineering ,Fractional programming ,Quadratic equation ,Conic section ,Modeling and Simulation ,0502 economics and business ,Convex optimization ,Bisection method ,Applied mathematics ,Quadratic programming ,Time complexity ,Mathematics - Abstract
Under the assumption that uncertain coefficients corresponding to each constraint are perturbed in an ellipsoidal set, we consider the problem of maximizing the perturbation radius of the ellipsoidal set associated to a robust convex quadratically constrained quadratic programming problem to maintain some properties of a pre-decision. To this end, a fractional programming problem is first formulated to solve the problem, and then equivalently reformulated into linear conic programs over positive semi-definite, second-order cones that are solvable in polynomial time. Numerical experiments in connection with the robust Markowitz’s portfolio selection problem are provided to demonstrate the proposed concept of sensitivity analysis. Additionally, certain numerical results are also presented to compare the efficiency of direct solutions of the proposed linear conic programs with that of a bisection method for the corresponding fractional programming problem.
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- 2021
41. Utility‐based pricing and hedging of contingent claims in Almgren‐Chriss model with temporary price impact
- Author
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Ibrahim Ekren and Sergey Nadtochiy
- Subjects
Quadratic growth ,Economics and Econometrics ,Applied Mathematics ,Probabilistic logic ,Hamilton–Jacobi–Bellman equation ,Indifference price ,Order (exchange) ,Accounting ,Bellman equation ,Economics ,Representation (mathematics) ,Asymptotic expansion ,Mathematical economics ,Social Sciences (miscellaneous) ,Finance - Abstract
In this paper, we construct the utility-based optimal hedging strategy for a European-type option in the Almgren-Chriss model with temporary price impact. The main mathematical challenge of this work stems from the degeneracy of the second order terms and the quadratic growth of the first order terms in the associated HJB equation, which makes it difficult to establish sufficient regularity of the value function needed to construct the optimal strategy in a feedback form. By combining the analytic and probabilistic tools for describing the value function and the optimal strategy, we establish the feedback representation of the latter. We use this representation to derive an explicit asymptotic expansion of the utility indifference price of the option, which allows us to quantify the price impact in options' market via the price impact coefficient in the underlying market.
- Published
- 2021
42. FGP approach to quadratically constrained multi-objective quadratic fractional programming with parametric functions
- Author
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Namrata Rani, Vandana Goyal, and Deepak Gupta
- Subjects
Quadratic growth ,Mathematical optimization ,Fractional programming ,Quadratic equation ,Optimization problem ,Computer science ,Management Science and Operations Research ,Decision maker ,Parametric equation ,Fuzzy goal programming ,Computer Science Applications ,Information Systems ,Management Information Systems - Abstract
This paper presents a quadratically constrained multiobjective quadratic fractional programming model (MOQFPM) and proposed a methodology to obtain a best preferred solution with the help of parametric functions and using fuzzy goal programming. In the initial stage, we obtain a non-fractional optimization model from the multi-objective quadratic fractional programming model by assigning a vector of parameters to fractional functions. Then, in the next stage, we use fuzzy goal programming approach to obtain the best preferred solution for the decision maker to the optimization problem by finding membership functions and aspiration levels of each objective function. This methodology proposes an efficient method to obtain Pareto-optimal solution of MOQFPM.
- Published
- 2021
43. Quadratic convergence analysis of a nonmonotone Levenberg–Marquardt type method for the weighted nonlinear complementarity problem
- Author
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Jingyong Tang and Jinchuan Zhou
- Subjects
Quadratic growth ,Levenberg–Marquardt algorithm ,Computational Mathematics ,Nonlinear system ,Control and Optimization ,Optimization problem ,Rate of convergence ,Applied Mathematics ,Complementarity (molecular biology) ,Applied mathematics ,Function (mathematics) ,Nonlinear complementarity problem ,Mathematics - Abstract
In this paper we consider the weighted nonlinear complementarity problem (denoted by wNCP) which contains a wide class of optimization problems. We introduce a family of new weighted complementarity functions and show that it is continuously differentiable everywhere and has several favorable properties. Based on this function, we reformulate the wNCP as a smooth nonlinear equation and propose a nonmonotone Levenberg–Marquardt type method to solve it. We show that the proposed method is well-defined and it is globally convergent without any additional condition. Moreover, we prove that the whole iteration sequence converges to a solution of the wNCP locally superlinearly or quadratically under the nonsingularity condition. In addition, we establish the local quadratic convergence of the proposed method under the local error bound condition. Some numerical results are also reported.
- Published
- 2021
44. The sufficient condition to find an exact dual bound in a separable quadratic optimization problem
- Author
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Oleg Berezovskyi
- Subjects
Quadratic growth ,Hessian matrix ,Semidefinite programming ,symbols.namesake ,Quadratic equation ,Duality gap ,Positive definiteness ,symbols ,Applied mathematics ,Relaxation (approximation) ,Quadratic programming ,Mathematics - Abstract
The paper considers nonconvex separable quadratic optimization problems subject to inequality constraints. A sufficient condition is given for finding the value and the point of the global extremum of a problem of this type by calculating the Lagrange dual bound. The peculiarity of this condition is that it is easily verified and requires from the Hessian matrix of the Lagrange function only that its region of positive definiteness is not empty. The result obtained for the dual bound also holds for the bound obtained using SDP relaxation. References Shor, N. Z., Stetsenko, S. I. (1989). Quadratic extremal problems and nondifferentiable optimization. Naukova Dumka, Kiev. Berezovskyi, O. A. (2017). Zero duality gap in quadratically constrained quadratic programming. Mathematical and computer modelling. Series: Physical and mathematical sciences, 15, 20-25. Nesterov, Y., Wolkowicz, H., Ye, Y. (2000). Semidefinite programming relaxations of nonconvex quadratic optimization. Handbook of semidefinite programming, Springer, New York, 361-419. DOI doi.org/10.1007/978-1-4615-4381-7_13 Berezovskyi, O. A. (2016). Exactness criteria for SDP-relaxations of quadratic extremum problems. Cybernetics and Systems Analysis, 52(6), 915-920. DOI doi.org/10.1007/s10559-016-9893-3
- Published
- 2021
45. Multiparametric/explicit nonlinear model predictive control for quadratically constrained problems
- Author
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Efstratios N. Pistikopoulos, Nikolaos A. Diangelakis, and Iosif Pappas
- Subjects
Quadratic growth ,0209 industrial biotechnology ,Mathematical optimization ,Computer science ,Stability (learning theory) ,02 engineering and technology ,Optimal control ,Industrial and Manufacturing Engineering ,Computer Science Applications ,symbols.namesake ,Model predictive control ,020901 industrial engineering & automation ,Quadratic equation ,020401 chemical engineering ,Control and Systems Engineering ,Modeling and Simulation ,Taylor series ,symbols ,Pruning (decision trees) ,Sensitivity (control systems) ,0204 chemical engineering - Abstract
Explicit model predictive control is an established methodology for the offline determination of the optimal control policy for linear discrete time-invariant systems with linear constraints. Nevertheless, nonlinearities in the form of quadratic constraints naturally appear in process models or are imposed for stability purposes in model predictive control formulations. In this manuscript, we present the theoretical developments and propose an algorithm for the exact solution of explicit nonlinear model predictive control problems with convex quadratic constraints. Our approach is based on a second-order Taylor approximation of Fiacco’s Basic Sensitivity Theorem, which allows for the existence and the analytic derivation of the optimal control actions. The complete exploration of the parameter space is founded on an active set strategy, which employs a pruning criterion to eliminate infeasible active sets. Based on that, the optimal map of solutions is constructed along with the corresponding control actions. The proposed strategy is applied to an explicit nonlinear model predictive control problem with an ellipsoidal terminal set, and comparisons with approximate solutions are drawn to demonstrate the benefits of the presented approach. Furthermore, as a practical application, the optimal operation of a chemostat in the presence of disturbances is exhibited.
- Published
- 2021
46. A multi-dimensional FBSDE with quadratic generator and its applications
- Author
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Rotenstein Eduard
- Subjects
forward-backward sdes ,quadratic growth ,financial derivatives ,Mathematics ,QA1-939 - Abstract
We consider, in the Markovian framework, a multi-dimensional forward - back - ward stochastic differential equation with quadratic growth for the generator function of the backward system. We prove an existence result of the solution and we use this result for pricing and hedging of contingent claims that depend on non-tradeable indexes by portfolios consisting in correlated risky assets.
- Published
- 2015
- Full Text
- View/download PDF
47. Summary
- Author
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Siegfried, Robert and Siegfried, Robert
- Published
- 2014
- Full Text
- View/download PDF
48. Results on Numerics for FBSDE with Drivers of Quadratic Growth
- Author
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Imkeller, Peter, Dos Reis, Gonçalo, Zhang, Jianing, Chiarella, Carl, editor, and Novikov, Alexander, editor
- Published
- 2010
- Full Text
- View/download PDF
49. Emergent Complexity in Conway’s Game of Life
- Author
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Gotts, Nick and Adamatzky, Andrew, editor
- Published
- 2010
- Full Text
- View/download PDF
50. Introduction
- Author
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Hetland, Magnus Lie and Hetland, Magnus Lie
- Published
- 2010
- Full Text
- View/download PDF
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