Improved Hessian estimation for adaptive random directions stochastic approximation
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Abstract:
We propose an improved Hessian estimation scheme for the second-order random directions stochastic approximation (2RDSA) algorithm [1]. The proposed scheme, inspired by [2], reduces the error in the Hessian estimate by (i) incorporating a zero-mean feedback term; and (ii) optimizing the step-sizes used in the Hessian recursion of 2RDSA.We prove that 2RDSA with our Hessian improvement scheme (2RDSA-IH) converges asymptotically to the true Hessian. The advantage with 2RDSA-IH is that it requires only 75% of the simulation cost per-iteration for 2SPSA with improved Hessian estimation (2SPSA-IH) [2]. Numerical experiments show that 2RDSA-IH outperforms both 2SPSA-IH and 2RDSA without the improved Hessian estimation scheme.Keywords:
Hessian matrix
Hessian equation
Stochastic Approximation
In this short note we present new integral formulas for the Hessian determinant. We use them for new definitions of Hessian under minimal regularity assumptions. The Hessian becomes a continuous linear functional on a Sobolev space.
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Sobolev inequality
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We show that the second–order condition for strict local extrema in both constrained and unconstrained optimization problems can be expressed solely in terms of principal minors of the (Lagrengean) Hessian. This approach unifies the determinantal tests in the sense that the second-order condition can be always given solely in terms of Hessian matrix.
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Hessian equation
Maxima and minima
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We derive Hessian estimates for convex solutions to quadratic Hessian equation by a compactness argument.
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Argument (complex analysis)
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The $k$-Hessian operator $\sigma_k$ is the $k$-th elementary symmetric function of the eigenvalues of the Hessian. It is known that the $k$-Hessian equation $\sigma_k(D^2u)=f$ with Dirichlet boundary condition $u=0$ is variational; indeed, this problem can be studied by means of the $k$-Hessian energy $-\int u\sigma_k(D^2u)$. We construct a natural boundary functional which, when added to the $k$-Hessian energy, yields as its critical points solutions of $k$-Hessian equations with general non-vanishing boundary data. As a consequence, we prove a sharp Sobolev trace inequality for $k$-admissible functions $u$ which estimates the $k$-Hessian energy in terms of the boundary values of $u$.
Hessian equation
Hessian matrix
TRACE (psycholinguistics)
Sobolev inequality
Sigma
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Hessian matrix
Hessian equation
Sequence (biology)
Quasi-Newton method
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Hessian matrix
Hessian equation
Unit sphere
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In this paper, we consider the following system coupled by multiparameter k ‐Hessian equations Here, λ 1 and λ 2 are positive parameters, Ω is the unit ball in R n , S k ( D 2 u ) is the k ‐Hessian operator of u , . Applying the eigenvalue theory in cones, several new results are obtained for the existence and multiplicity of nontrivial radial solutions for the above k ‐Hessian system. In particular, we study the dependence of the nontrivial radial solution on the parameter λ 1 and λ 2 . This is probably the first time that a system of equations, especially with fully nonlinear equations, has been studied by applying this technique. Finally, as an application, we obtain sufficient conditions for the existence of nontrivial radial solutions of the power‐type coupled system of k ‐Hessian equations, which is new even for the special case k =1 and extends a previous result for the case k = n .
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Hessian matrix
Multiplicity (mathematics)
Unit sphere
Ball (mathematics)
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Hessian matrix
Hessian equation
Dirichlet eigenvalue
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The $k$-Hessian operator $\sigma_k$ is the $k$-th elementary symmetric function of the eigenvalues of the Hessian. It is known that the $k$-Hessian equation $\sigma_k(D^2u)=f$ with Dirichlet boundary condition $u=0$ is variational; indeed, this problem can be studied by means of the $k$-Hessian energy $-\int u\sigma_k(D^2u)$. We construct a natural boundary functional which, when added to the $k$-Hessian energy, yields as its critical points solutions of $k$-Hessian equations with general non-vanishing boundary data. As a consequence, we prove a sharp Sobolev trace inequality for $k$-admissible functions $u$ which estimates the $k$-Hessian energy in terms of the boundary values of $u$.
Hessian equation
Hessian matrix
TRACE (psycholinguistics)
Sobolev inequality
Sigma
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