Lipschitzian optimization without the Lipschitz constant

1993 ◽  
Vol 79 (1) ◽  
pp. 157-181 ◽  
Author(s):  
D. R. Jones ◽  
C. D. Perttunen ◽  
B. E. Stuckman
Keyword(s):  
Lipschitz Constant ◽  
Lipschitzian Optimization

scholarly journals On the experimental investigation of Pareto–Lipschitzian optimization

2011 ◽  
Vol 52 ◽  
Author(s):  
Jonas Mockus ◽  
Justas Stašionis
Keyword(s):  
Experimental Investigation ◽  
Global Optimization ◽  
Pareto Optimality ◽  
Lipschitz Constant ◽  
Real Life ◽  
Multiple Criteria ◽  
Life Problems ◽  
Lipschitzian Optimization ◽  
Lipschitz Constants ◽  
Novel Method

A well-known example of global optimization that provides solutions within fixed error limits is optimization of functions with a known Lipschitz constant. In many real-life problems this constant is unknown. To address that, we propose a novel method called Pareto Lipschitzian Optimization (PLO) that provides solutions within fixed error limits for functions with unknown Lipschitzconstants.In the proposed approach, a set of all unknown Lipschitz constants is regarded as multiple criteria using the concept of Pareto Optimality (PO).

The least distance between extremums and minimal period of solutions to autonomous vector differential equations

2019 ◽  
Vol 485 (2) ◽  
pp. 142-144
Author(s):  
A. A. Zevin
Keyword(s):  
Lower Bound ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Lipschitz Constant ◽  
Minimal Period ◽  
Arbitrary Vector ◽  
Vector Norm ◽  
Sharp Lower Bound

Solutions x(t) of the Lipschitz equation x = f(x) with an arbitrary vector norm are considered. It is proved that the sharp lower bound for the distances between successive extremums of xk(t) equals π/L where L is the Lipschitz constant. For non-constant periodic solutions, the lower bound for the periods is 2π/L. These estimates are achieved for norms that are invariant with respect to permutation of the indices.

Trend to spatial homogeneity for solutions to semilinear damped wave equations

1987 ◽  
Vol 105 (1) ◽  
pp. 117-126 ◽  
Author(s):  
J. Solà-Morales ◽  
M. València
Keyword(s):  
Boundary Conditions ◽  
Wave Equations ◽  
Lipschitz Constant ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
Spatial Homogeneity ◽  
Homogeneous Solutions ◽  
Spatially Homogeneous ◽  
Image Position

SynopsisThe semilinear damped wave equationssubject to homogeneous Neumann boundary conditions, admit spatially homogeneous solutions (i.e. u(x, t) = u(t)). In order that every solution tends to a spatially homogeneous one, we look for conditions on the coefficients a and d, and on the Lipschitz constant of f with respect to u.

scholarly journals Region of existence of multiple solutions for a class of Robin type four-point BVPs

2021 ◽  
Vol 41 (4) ◽  
pp. 571-600
Author(s):  
Amit K. Verma ◽  
Nazia Urus ◽  
Ravi P. Agarwal
Keyword(s):  
Multiple Solutions ◽  
Nonlinear Boundary ◽  
Source Term ◽  
Lipschitz Constant ◽  
Sufficient Conditions ◽  
Existence Of Solution ◽  
Monotone Iterative Technique ◽  
Nonlinear Boundary Value Problems ◽  
Existence Of A Solution ◽  
Monotone Iterative

This article aims to prove the existence of a solution and compute the region of existence for a class of four-point nonlinear boundary value problems (NLBVPs) defined as \[\begin{gathered} -u''(x)=\psi(x,u,u'), \quad x\in (0,1),\\ u'(0)=\lambda_{1}u(\xi), \quad u'(1)=\lambda_{2} u(\eta),\end{gathered}\] where \(I=[0,1]\), \(0\lt\xi\leq\eta\lt 1\) and \(\lambda_1,\lambda_2\gt 0\). The nonlinear source term \(\psi\in C(I\times\mathbb{R}^2,\mathbb{R})\) is one sided Lipschitz in \(u\) with Lipschitz constant \(L_1\) and Lipschitz in \(u'\) such that \(|\psi(x,u,u')-\psi(x,u,v')|\leq L_2(x)|u'-v'|\). We develop monotone iterative technique (MI-technique) in both well ordered and reverse ordered cases. We prove maximum, anti-maximum principle under certain assumptions and use it to show the monotonic behaviour of the sequences of upper-lower solutions. The sufficient conditions are derived for the existence of solution and verified for two examples. The above NLBVPs is linearised using Newton's quasilinearization method which involves a parameter \(k\) equivalent to \(\max_u\frac{\partial \psi}{\partial u}\). We compute the range of \(k\) for which iterative sequences are convergent.

scholarly journals Elliptic Theory for Sets with Higher Co-dimensional Boundaries

2021 ◽  
Vol 274 (1346) ◽  
Author(s):  
G. David ◽  
J. Feneuil ◽  
S. Mayboroda
Keyword(s):  
Green Function ◽  
Elliptic Operator ◽  
Harmonic Measure ◽  
Lipschitz Constant ◽  
Extension Theorems ◽  
Content Type ◽  
Elliptic Theory ◽  
Basic Set ◽  
Degenerate Elliptic ◽  
The Green Function

Many geometric and analytic properties of sets hinge on the properties of elliptic measure, notoriously missing for sets of higher co-dimension. The aim of this manuscript is to develop a version of elliptic theory, associated to a linear PDE, which ultimately yields a notion analogous to that of the harmonic measure, for sets of codimension higher than 1. To this end, we turn to degenerate elliptic equations. Let Γ ⊂ R n \Gamma \subset \mathbb {R}^n be an Ahlfors regular set of dimension d > n − 1 d>n-1 (not necessarily integer) and Ω = R n ∖ Γ \Omega = \mathbb {R}^n \setminus \Gamma . Let L = − div ⁡ A ∇ L = - \operatorname {div} A\nabla be a degenerate elliptic operator with measurable coefficients such that the ellipticity constants of the matrix A A are bounded from above and below by a multiple of dist ⁡ ( ⋅ , Γ ) d + 1 − n \operatorname {dist}(\cdot , \Gamma )^{d+1-n} . We define weak solutions; prove trace and extension theorems in suitable weighted Sobolev spaces; establish the maximum principle, De Giorgi-Nash-Moser estimates, the Harnack inequality, the Hölder continuity of solutions (inside and at the boundary). We define the Green function and provide the basic set of pointwise and/or L p L^p estimates for the Green function and for its gradient. With this at hand, we define harmonic measure associated to L L , establish its doubling property, non-degeneracy, change-of-the-pole formulas, and, finally, the comparison principle for local solutions. In another article to appear, we will prove that when Γ \Gamma is the graph of a Lipschitz function with small Lipschitz constant, we can find an elliptic operator L L for which the harmonic measure given here is absolutely continuous with respect to the d d -Hausdorff measure on Γ \Gamma and vice versa. It thus extends Dahlberg’s theorem to some sets of codimension higher than 1.

Lipschitz continuity of the Fréchet gradient in an inverse coefficient problem for a parabolic equation with Dirichlet measured output

2018 ◽  
Vol 26 (3) ◽  
pp. 349-368 ◽  
Author(s):  
Alemdar Hasanov
Keyword(s):  
Parabolic Equation ◽  
Lipschitz Continuity ◽  
Lipschitz Constant ◽  
Sufficient Conditions ◽  
Gradient Methods ◽  
Neumann Boundary ◽  
Iteration Algorithm ◽  
Coefficient Problem ◽  
Inverse Coefficient Problem ◽  
Inverse Coefficient

AbstractThis paper studies the Lipschitz continuity of the Fréchet gradient of the Tikhonov functional {J(k):=(1/2)\lVert u(0,\cdot\,;k)-f\rVert^{2}_{L^{2}(0,T)}} corresponding to an inverse coefficient problem for the {1D} parabolic equation {u_{t}=(k(x)u_{x})_{x}} with the Neumann boundary conditions {-k(0)u_{x}(0,t)=g(t)} and {u_{x}(l,t)=0}. In addition, compactness and Lipschitz continuity of the input-output operator\Phi[k]:=u(x,t;k)\lvert_{x=0^{+}},\quad\Phi[\,\cdot\,]:\mathcal{K}\subset H^{1% }(0,l)\mapsto H^{1}(0,T),as well as solvability of the regularized inverse problem and the Lipschitz continuity of the Fréchet gradient of the Tikhonov functional are proved. Furthermore, relationships between the sufficient conditions for the Lipschitz continuity of the Fréchet gradient and the regularity of the weak solution of the direct problem as well as the measured output {f(t):=u(0,t;k)} are established. One of the derived lemmas also introduces a useful application of the Lipschitz continuity of the Fréchet gradient. This lemma shows that an important advantage of gradient methods comes when dealing with the functionals of class {C^{1,1}(\mathcal{K})}. Specifically, this lemma asserts that if {J\in C^{1,1}(\mathcal{K})} and {\{k^{(n)}\}\subset\mathcal{K}} is the sequence of iterations obtained by the Landweber iteration algorithm {k^{(n+1)}=k^{(n)}+\omega_{n}J^{\prime}(k^{(n)})}, then for {\omega_{n}\in(0,2/L_{g})}, where {L_{g}>0} is the Lipschitz constant, the sequence {\{J(k^{(n)})\}} is monotonically decreasing and {\lim_{n\to\infty}\lVert J^{\prime}(k^{(n)})\rVert=0}.

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