A Strongly Semismooth Integral Function and Its Application
20 Mar 2003-Computational Optimization and Applications (Kluwer Academic Publishers)-Vol. 25, Iss: 1, pp 223-246
TL;DR: It is shown that f is a strongly semismooth function if g is continuous and B is affine with respect to t and stronglySemismooth withrespect to x, i.e., B(x, t) = u(x)t + v(x), where u and v are two strongly Semismooth functions in ℝn.
Abstract: As shown by an example, the integral function f : {\bb R}n → {\bb R}, defined by f(x) e ∫ab[B(x, t)]+g(t) dt, may not be a strongly semismooth function, even if g(t) ≡ 1 and B is a quadratic polynomial with respect to t and infinitely many times smooth with respect to x. We show that f is a strongly semismooth function if g is continuous and B is affine with respect to t and strongly semismooth with respect to x, i.e., B(x, t) e u(x)t + v(x), where u and v are two strongly semismooth functions in {\bb R}n. We also show that f is not a piecewise smooth function if u and v are two linearly independent linear functions, g is continuous and g n 0 in [a, b], and n ≥ 2. We apply the first result to the edge convex minimum norm network interpolation problem, which is a two-dimensional interpolation problem.
Citations
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TL;DR: The tensor absolute value equations (TVEE) as mentioned in this paper is a generalization of the well-known tensor complementarity problems in the matrix case and is related to tensor complementary problems.
Abstract: This paper is concerned with solving some structured multi-linear systems, which are called tensor absolute value equations This kind of absolute value equations is closely related to tensor complementarity problems and is a generalization of the well-known absolute value equations in the matrix case We prove that tensor absolute value equations are equivalent to some special structured tensor complementary problems Some sufficient conditions are given to guarantee the existence of solutions for tensor absolute value equations We also propose a Levenberg-Marquardt-type algorithm for solving some given tensor absolute value equations and preliminary numerical results are reported to indicate the efficiency of the proposed algorithm
44 citations
TL;DR: This note brings out a property of the functions that enter such equations, for instance through penalty expressions, that is increasingly important in the numerical treatment of complementarity problems and models of equilibrium.
Abstract: Piecewise smooth equations are increasingly important in the numerical treatment of complementarity problems and models of equilibrium. This note brings out a property of the functions that enter such equations, for instance through penalty expressions.
42 citations
Cites background from "A Strongly Semismooth Integral Func..."
...Basic background and developments in the subject can be found in [1–6, 8, 9], and more recently for example in [7]....
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TL;DR: This paper study differentiability and semismoothness properties of functions defined as integrals of parameterized functions and applications of the developed theory to the problems of shape-preserving interpolation, option pricing and semi-infinite programming.
Abstract: In this paper we study differentiability and semismoothness properties of functions defined as integrals of parameterized functions. We also discuss applications of the developed theory to the problems of shape-preserving interpolation, option pricing and semi-infinite programming.
30 citations
Cites background or methods from "A Strongly Semismooth Integral Func..."
...An example of an integral function F(·), of the form (2.9), which is not strongly semismooth was given in Qi and Yin [20]....
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...1) were discussed in recent publications by Dontchev, Qi and Qi [4],[5], Qi [16], Qi and Tseng [19], Qi and Yin [20], and Wang, Yin and Qi [33]....
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...Wang, Yin and Qi [33] developed an interpolation method to preserve the shape of the option price function....
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...9), which is not strongly semismooth was given in Qi and Yin [20]....
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...In [20], this result was generalized to a class of integral functions, which are still a special case of (1....
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Posted Content•
TL;DR: In this paper, the authors prove that tensor absolute value equations are equivalent to some special structured tensor complementary problems and propose a Levenberg-Marquardt-type algorithm for solving these problems.
Abstract: This paper is concerned with solving some structured multi-linear systems, which are called tensor absolute value equations. This kind of absolute value equations is closely related to tensor complementarity problems and is a generalization of the well-known absolute value equations in the matrix case. We prove that tensor absolute value equations are equivalent to some special structured tensor complementary problems. Some sufficient conditions are given to guarantee the existence of solutions for tensor absolute value equations. We also propose a Levenberg-Marquardt-type algorithm for solving some given tensor absolute value equations and preliminary numerical results are reported to indicate the efficiency of the proposed algorithm.
27 citations
TL;DR: In this article, an interpolation method is developed to preserve the shape of the option price function, which is optimal in terms of minimizing the distance between the implied risk-neutral density and the prior approximation function in L 2-norm.
Abstract: Several risk management and exotic option pricing models have been proposed in the literature which may price European options correctly. A prerequisite of these models is the interpolation of the market implied volatilities or the European option price function. However, the no-arbitrage principle places shape restrictions on the option price function. In this paper, an interpolation method is developed to preserve the shape of the option price function. The interpolation is optimal in terms of minimizing the distance between the implied risk-neutral density and the prior approximation function in L
2-norm, which is important when only a few observations are available. We reformulate the problem into a system of semismooth equations so that it can be solved efficiently.
26 citations
References
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Book•
01 Jan 1983
TL;DR: The Calculus of Variations as discussed by the authors is a generalization of the calculus of variations, which is used in many aspects of analysis, such as generalized gradient descent and optimal control.
Abstract: 1. Introduction and Preview 2. Generalized Gradients 3. Differential Inclusions 4. The Calculus of Variations 5. Optimal Control 6. Mathematical Programming 7. Topics in Analysis.
9,498 citations
TL;DR: It is shown that the gradient function of the augmented Lagrangian forC2-nonlinear programming is semismooth, and the extended Newton's method can be used in the augmentedlagrangian method for solving nonlinear programs.
Abstract: Newton's method for solving a nonlinear equation of several variables is extended to a nonsmooth case by using the generalized Jacobian instead of the derivative. This extension includes the B-derivative version of Newton's method as a special case. Convergence theorems are proved under the condition of semismoothness. It is shown that the gradient function of the augmented Lagrangian forC2-nonlinear programming is semismooth. Thus, the extended Newton's method can be used in the augmented Lagrangian method for solving nonlinear programs.
1,464 citations
"A Strongly Semismooth Integral Func..." refers background or methods in this paper
...Semismooth functions, as introduced by Mifflin [21] for functionals and further extended by Qi and Sun [ 26 ] for vector-valued functions, play an important role in superlinear convergence analysis of generalized Newton methods for solving nonsmooth equations....
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...The following lemma is extracted from Theorem 2.3 of [ 26 ]....
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...As proved in [17, 24, 26 ], the generalized Newton method has local superlinear (quadratic) convergence provided that the function is (strongly) semismooth and the matrices in the Clarke...
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...the k-th step of the generalized (semismooth) Newton method, as considered in [ 26 ], has the following form: Choose Vk ∈ ∂ F(xk), and find...
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...It has been proved in [ 26 ] that F is semismooth at x if and only if all its component functions are semismooth....
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TL;DR: In this paper, the authors introduce semismooth and semiconvex functions and discuss their properties with respect to nonsmooth nonconvex constrained optimization problems and give a chain rule for generalized gradients.
Abstract: We introduce semismooth and semiconvex functions and discuss their properties with respect to nonsmooth nonconvex constrained optimization problems. These functions are locally Lipschitz, and hence have generalized gradients. The author has given an optimization algorithm that uses generalized gradients of the problem functions and converges to stationary points if the functions are semismooth. If the functions are semiconvex and a constraint qualification is satisfied, then we show that a stationary point is an optimal point.
We show that the pointwise maximum or minimum over a compact family of continuously differentiable functions is a semismooth function and that the pointwise maximum over a compact family of semiconvex functions is a semiconvex function. Furthermore, we show that a semismooth composition of semismooth functions is semismooth and gives a type of chain rule for generalized gradients.
830 citations
"A Strongly Semismooth Integral Func..." refers background in this paper
...Semismooth functions, as introduced by Mifflin [ 21 ] for functionals and further extended by Qi and Sun [26] for vector-valued functions, play an important role in superlinear convergence analysis of generalized Newton methods for solving nonsmooth equations....
[...]
TL;DR: Convergence analysis of some algorithms for solving systems of nonlinear equations defined by locally Lipschitzian functions and a hybrid method, which is both globally convergent in the sense of finding a stationary point of the norm function of the system and locally quadratically convergent, is presented.
Abstract: This paper presents convergence analysis of some algorithms for solving systems of nonlinear equations defined by locally Lipschitzian functions. For the directional derivative-based and the generalized Jacobian-based Newton methods, both the iterates and the corresponding function values are locally, superlinearly convergent. Globally, a limiting point of the iterate sequence generated by the damped, directional derivative-based Newton method is a zero of the system if and only if the iterate sequence converges to this point and the stepsize eventually becomes one, provided that the system is strongly BD-regular and semismooth at this point. In this case, the convergence is superlinear. A general attraction theorem is presented, which can be applied to two algorithms proposed by Han, Pang and Rangaraj. A hybrid method, which is both globally convergent in the sense of finding a stationary point of the norm function of the system and locally quadratically convergent, is also presented.
727 citations
"A Strongly Semismooth Integral Func..." refers background or methods in this paper
...As proved in [17, 24 , 26], the generalized Newton method has local superlinear (quadratic) convergence provided that the function is (strongly) semismooth and the matrices in the Clarke...
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...[ 24 , 26] different names for strong semismoothness were used.) A function F is said to be STRONGLY SEMISMOOTH INTEGRAL FUNCTION AND ITS APPLICATION 227...
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...Then the B-subdifferential of F at x [ 24 ] is defined to be...
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TL;DR: It is proved that three most often used Gabriel-Moré smoothing functions can generate strongly semismooth functions, which play a fundamental role in establishing superlinear and quadratic convergence of the new smoothing Newton methods.
Abstract: In this paper we take a new look at smoothing Newton methods for solving the nonlinear complementarity problem (NCP) and the box constrained variational inequalities (BVI). Instead of using an infinite sequence of smoothing approximation functions, we use a single smoothing approximation function and Robinson’s normal equation to reformulate NCP and BVI as an equivalent nonsmooth equation H(u,x)=0, where H:ℜ
2n
→ℜ
2n
, u∈ℜ
n
is a parameter variable and x∈ℜ
n
is the original variable. The central idea of our smoothing Newton methods is that we construct a sequence {z
k
=(u
k
,x
k
)} such that the mapping H(·) is continuously differentiable at each z
k
and may be non-differentiable at the limiting point of {z
k
}. We prove that three most often used Gabriel-More smoothing functions can generate strongly semismooth functions, which play a fundamental role in establishing superlinear and quadratic convergence of our new smoothing Newton methods. We do not require any function value of F or its derivative value outside the feasible region while at each step we only solve a linear system of equations and if we choose a certain smoothing function only a reduced form needs to be solved. Preliminary numerical results show that the proposed methods for particularly chosen smoothing functions are very promising.
360 citations
"A Strongly Semismooth Integral Func..." refers background in this paper
...This is guaranteed by the semismoothness (strong semismoothness) of the reformulation equations, see [4, 12, 14, 18, 25, 27 , 35] and references therein for example....
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