• Title, Summary, Keyword: approximate convexity

### APPROXIMATE CONVEXITY WITH RESPECT TO INTEGRAL ARITHMETIC MEAN

• Zoldak, Marek
• Bulletin of the Korean Mathematical Society
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• v.51 no.6
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• pp.1829-1839
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• 2014
• Let (${\Omega}$, $\mathcal{S}$, ${\mu}$) be a probabilistic measure space, ${\varepsilon}{\in}\mathbb{R}$, ${\delta}{\geq}0$, p > 0 be given numbers and let $P{\subset}\mathbb{R}$ be an open interval. We consider a class of functions $f:P{\rightarrow}\mathbb{R}$, satisfying the inequality $$f(EX){\leq}E(f{\circ}X)+{\varepsilon}E({\mid}X-EX{\mid}^p)+{\delta}$$ for each $\mathcal{S}$-measurable simple function $X:{\Omega}{\rightarrow}P$. We show that if additionally the set of values of ${\mu}$ is equal to [0, 1] then $f:P{\rightarrow}\mathbb{R}$ satisfies the above condition if and only if $$f(tx+(1-t)y){\leq}tf(x)+(1-t)f(y)+{\varepsilon}[(1-t)^pt+t^p(1-t)]{\mid}x-y{\mid}^p+{\delta}$$ for $x,y{\in}P$, $t{\in}[0,1]$. We also prove some basic properties of such functions, e.g. the existence of subdifferentials, Hermite-Hadamard inequality.

### Sequential Approximate Optimization by Dual Method Based on Two-Point Diagonal Quadratic Approximation (이점 대각 이차 근사화 기법을 쌍대기법에 적용한 순차적 근사 최적설계)

• Park, Seon-Ho;Jung, Sang-Jin;Jeong, Seung-Hyun;Choi, Dong-Hoon
• Transactions of the Korean Society of Mechanical Engineers A
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• v.35 no.3
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• pp.259-266
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• 2011
• We present a new dual sequential approximate optimization (SAO) algorithm called SD-TDQAO (sequential dual two-point diagonal quadratic approximate optimization). This algorithm solves engineering optimization problems with a nonlinear objective and nonlinear inequality constraints. The two-point diagonal quadratic approximation (TDQA) was originally non-convex and inseparable quadratic approximation in the primal design variable space. To use the dual method, SD-TDQAO uses diagonal quadratic explicit separable approximation; this can easily ensure convexity and separability. An important feature is that the second-derivative terms of the quadratic approximation are approximated by TDQA, which uses only information on the function and the derivative values at two consecutive iteration points. The algorithm will be illustrated using mathematical and topological test problems, and its performance will be compared with that of the MMA algorithm.

### APPROXIMATING FIXED POINTS OF NONEXPANSIVE TYPE MAPPINGS IN BANACH SPACES WITHOUT UNIFORM CONVEXITY

• Sahu, Daya Ram;Khan, Abdul Rahim;Kang, Shin Min
• Bulletin of the Korean Mathematical Society
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• v.50 no.3
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• pp.1007-1020
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• 2013
• Approximate fixed point property problem for Mann iteration sequence of a nonexpansive mapping has been resolved on a Banach space independent of uniform (strict) convexity by Ishikawa [Fixed points and iteration of a nonexpansive mapping in a Banach space, Proc. Amer. Math. Soc. 59 (1976), 65-71]. In this paper, we solve this problem for a class of mappings wider than the class of asymptotically nonexpansive mappings on an arbitrary normed space. Our results generalize and extend several known results.

### MONOTONE ITERATION SCHEME FOR IMPULSIVE THREE-POINT NONLINEAR BOUNDARY VALUE PROBLEMS WITH QUADRATIC CONVERGENCE

• Journal of the Korean Mathematical Society
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• v.45 no.5
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• pp.1275-1295
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• 2008
• In this paper, we consider an impulsive nonlinear second order ordinary differential equation with nonlinear three-point boundary conditions and develop a monotone iteration scheme by relaxing the convexity assumption on the function involved in the differential equation and the concavity assumption on nonlinearities in the boundary conditions. In fact, we obtain monotone sequences of iterates (approximate solutions) converging quadratically to the unique solution of the impulsive three-point boundary value problem.

### Geometric Processing for Freeform Surfaces Based on High-Precision Torus Patch Approximation (토러스 패치 기반의 정밀 근사를 이용한 자유곡면의 기하학적 처리)

• Park, Youngjin;Hong, Q Youn;Kim, Myung-Soo
• Journal of The Korea Computer Graphics Society
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• v.25 no.3
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• pp.93-103
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• 2019
• We introduce a geometric processing method for freeform surfaces based on high-precision torus patch approximation, a new spatial data structure for efficient geometric operations on freeform surfaces. A torus patch fits the freeform surface with flexibility: it can handle not only positive and negative curvature but also a zero curvature. It is possible to precisely approximate the surface regardless of the convexity/concavity of the surface. Unlike the traditional method, a torus patch easily bounds the surface normal, and the offset of the torus becomes a torus again, thus helps the acceleration of various geometric operations. We have shown that the torus patch's approximation accuracy of the freeform surface is high by measuring the upper bound of the two-sided Hausdorff distance between the freeform surface and set of torus patches. Using the method, it can be easily processed to detect an intersection curve between two freeform surfaces and find the offset surface of the freeform surface.