RETRACTED: Double well potential function and its optimization in the <i>n</i>-dimensional real space: part I

Song, Fang; Gao, David Yang; Lin, Gang Xuan; Sheu, Ruey-Lin; Xing, Wen

doi:10.1177/1081286514566704

Cited by 9 publications

(13 citation statements)

References 12 publications

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“…A similar argument applies for any other components. h The next result shows that the sufficient condition 1 2 k w Ã k 2 À n + a 1 ! 0 in Lemma 3 is indeed necessary for a critical point to become the global minimizer.…”

Section: Characterization Of the Global Minimizermentioning

confidence: 84%

“…For simplicity, we define n = À 1 2 c T (I À B(B T B) À1 B T )c + d and c = P T f À DP T B T c. By dropping the constant terms, we can rewrite problem (DWP) defined in (1) Recall that the canonical primal problem defined in (19) of part I is to minimize…”

Section: Characterization Of a Local Nonglobal Minimizermentioning

confidence: 99%

“…We remark that characterizing the local maximizer of the trust-region subproblem (4)-( 5) can be reduced to the problem of finding a local minimizer of (4) with A being replaced by ÀA. However, due to the nonsymmetric nature, it is no longer the case for the double well potential problem (1). Hence Martı´nez's approach may not be able to characterize the local maximizer for a general (DWP) problem.…”

Section: Introductionmentioning

confidence: 99%

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RETRACTED: Double well potential function and its optimization in the n-dimensional real space: part II

Xia

Sheu

Fang

et al. 2016

Mathematics and Mechanics of Solids

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The following article has been included in a multiple retraction: Double Well Potential Function and Its Optimization in The n-dimensional Real Space. Part II; Yong Xia, Ruey-Lin Sheu, Shu-Cherng Fang, Wenxun Xing. http://mms.sagepub.com/content/early/2015/02/09/1081286514566723.abstract In 2015 SAGE were made aware of concerns regarding the Special Issue of Mathematics & Mechanics of Solids on Advances in Canonical Duality Theory, guest-edited by Professor David Gao. At the request of the Guest Editor, the Special Issue has been retracted, due to conflict of interest regarding Professor Gao’s role as Guest Editor and co-author on a number of submitted papers. In addition the peer review process was less rigorous than the journal requires. The Guest Editor takes full responsibility for the retraction. The following articles that were due to appear in the Special Issue have therefore been retracted: Canonical Duality-Triality: Bridge between Nonconvex Analysis/Mechanics and Global Optimization in complex systems; David Y Gao, Ning Ruan, and Vittorio Latorre. http://mms.sagepub.com/content/early/2015/02/24/1081286514566533.abstract Canonical Dual Approach for Contact Mechanics Problems with Friction; Vittorio Latorre, Simone Sagratella, David Y Gao. http://mms.sagepub.com/content/early/2015/01/20/1081286514566534.abstract Canonical Duality Theory for Solving Non-Monotone Variational Inequality Problems; Guoshan Liu, David Y Gao, Shouyang Wang. http://mms.sagepub.com/content/early/2015/02/04/1081286514566535.abstract Double Well Potential Function and Its Optimization in The n-dimensional Real Space. Part I; Shu-Cherng Fang, David Y Gao, Gang-Xuan Lin, Ruey-Lin Sheu, Wen-Xun Xing. http://mms.sagepub.com/content/early/2015/02/24/1081286514566704.abstract Double Well Potential Function and Its Optimization in The n-dimensional Real Space. Part II; Yong Xia, Ruey-Lin Sheu, Shu-Cherng Fang, Wenxun Xing. http://mms.sagepub.com/content/early/2015/02/09/1081286514566723.abstract Analytic Solutions to 3-D Finite Deformation Problems Governed by St Venant-Kirchhoff Material; David Y Gao and E. Hajilarov. http://mms.sagepub.com/content/early/2015/07/06/1081286515591084.abstract Triality Theory and Complete Post-buckling Solutions of Large Deformed Beam by Canonical Dual Finite Element Method; Kun Cai, David Y Gao, Qinghua Qin. http://mms.sagepub.com/content/early/2015/06/28/1081286515591085.abstract Global Solutions to Spherically Constrained Quadratic Minimization via Canonical Duality Theory; Yi Chen, David Y Gao. http://mms.sagepub.com/content/early/2015/04/08/1081286515577122.abstract Unified Canonical Duality Methodology for Global Optimization; Vittorio Latorre, David Y Gao and N. Ruan. http://mms.sagepub.com/content/early/2015/07/06/1081286515591305.abstract A Framework of Canonical Dual Algorithms for Global Optimization; Xiaojun Zhou, David Y Gao, Chunhua Yang. http://mms.sagepub.com/content/early/2015/07/22/1081286515592190.abstract Canonical Duality Theory for Solving Nonconvex/Discrete Constrained Globa...

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Section: Characterization Of the Global Minimizermentioning

confidence: 84%

Section: Characterization Of a Local Nonglobal Minimizermentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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RETRACTED: Double well potential function and its optimization in the n-dimensional real space: part II

Xia

Sheu

Fang

et al. 2016

Mathematics and Mechanics of Solids

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“…The special case (3-RS) is the wellknown Nestrov-Polyak subproblem [21], which is an important subproblem in regularized Newton methods for unconstrained optimization. The double-well potential minimization problem [22] corresponds to (4-RS). If A 0, h(x) is convex as p > 2.…”

Section: Generalized P-regularized Subproblem With P >mentioning

confidence: 99%

Unifying Farkas lemma and S-lemma: new theory and applications in nonquadratic nonconvex optimization

Yang¹,

Xia²,

Wang³

2021

Preprint

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“…As a consequence, both (QP1EQC) and (GTRS) are completely analyzed. We remark that (QP1EQC) itself has many interesting applications, including the double well potential optimization problem [10,35], the time of arrival geolocation problem [15] and unbiased least squares optimization for system identification [26]. In particular, the double well potential model came from numerical approximations to the generalized Ginzburg-Landau functionals [18].…”

Section: S-condition 2([3] Thm A2)mentioning

confidence: 99%

S-lemma with equality and its applications

2015

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Let f (x) = x T Ax + 2a T x + c and h(x) = x T Bx + 2b T x + d be two quadratic functions having symmetric matrices A and B. The S-lemma with equality asks when the unsolvability of the system f (x) < 0, h(x) = 0 implies the existence of a real number µ such that f (x) + µh(x) ≥ 0, ∀x ∈ R n . The problem is much harder than the inequality version which asserts that, under Slater condition, f (x) < 0, h(x) ≤ 0 is unsolvable if and only if f (x) + µh(x) ≥ 0, ∀x ∈ R n for some µ ≥ 0. In this paper, we show that the S-lemma with equality does not hold only when the matrix A has exactly one negative eigenvalue and h(x) is a non-constant linear function (B = 0, b = 0). As an application, we can globally solve inf{f (x) : h(x) = 0} as well as the two-sided generalized trust region subproblem inf{f (x) : l ≤ h(x) ≤ u} without any condition. Moreover, the convexity of the joint numerical range {(f (x), h 1 (x), . . . , h p (x)) : x ∈ R n } where f is a (possibly nonconvex) quadratic function and h 1 (x), . . . , h p (x) are affine functions can be characterized using the newly developed S-lemma with equality.

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RETRACTED: Double well potential function and its optimization in the n-dimensional real space: part I

Cited by 9 publications

References 12 publications

RETRACTED: Double well potential function and its optimization in the n-dimensional real space: part II

RETRACTED: Double well potential function and its optimization in the n-dimensional real space: part II

Unifying Farkas lemma and S-lemma: new theory and applications in nonquadratic nonconvex optimization

S-lemma with equality and its applications

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