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

Xia, Yong; Sheu, Ruey-Lin; Fang, Shu‐Cherng; Xing, Wenxun

doi:10.1177/1081286514566723

Cited by 5 publications

(8 citation statements)

References 56 publications

(118 reference statements)

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“…Our paper characterizes (p-RS) completely for any p > 2 by extending (i) the necessary and sufficient global optimality conditions for p = 3 in [2]; (ii) the analysis using the secular function (to be specified later) for p = 4 in [18]; and (iii) a necessary global optimality condition for p > 2 in [7]. Some generalization is, nevertheless, non-trivial in mathematical skills.…”

Section: Mathematics Subject Classification (2010) 49k30 90c46 90c26 ...mentioning

confidence: 98%

“…See Nesterov and Polyak [15]; Weiser Deuflhard and Erdmann [17]; and Cartis, Gould and Toint [2]. When p = 4, (p-RS) reduces to a form of the double well potential function which has many applications in solid mechanics and quantum mechanics [5,18]. Gould, Robinson and Thorne [7] studied (p-RS) for a general p > 2 in comparison with the the trust-region subproblem (TRS) min 1 2…”

Section: Mathematics Subject Classification (2010) 49k30 90c46 90c26 ...mentioning

confidence: 99%

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On the p-regularized trust region subproblem

Hsia,

Sheu,

Yuan

2014

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The p-regularized subproblem (p-RS) is a regularisation technique in computing a Newton-like step for unconstrained optimization, which globally minimizes a local quadratic approximation of the objective function while incorporating with a weighted regularisation term σ p x p . The global solution of the p-regularized subproblem for p = 3, also known as the cubic regularization, has been characterized in literature. In this paper, we resolve both the global and the local non-global minimizers of (p-RS) for p > 2 with necessary and sufficient optimality conditions. Moreover, we prove a parallel result of Martínez [13] that the (p-RS) for p > 2, analogous to the trust region subproblem, can have at most one local non-global minimizer. When the (p-RS) is subject to a fixed number m additional linear inequality constraints, we show that the uniqueness of the local solution of the (p-RS) (if exists at all), especially for p = 4, can be applied to solve such an extension in polynomial time.

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Section: Mathematics Subject Classification (2010) 49k30 90c46 90c26 ...mentioning

confidence: 98%

Section: Mathematics Subject Classification (2010) 49k30 90c46 90c26 ...mentioning

confidence: 99%

On the p-regularized trust region subproblem

Hsia,

Sheu,

Yuan

2014

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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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“…For more references, we refer to [17,18,19,20] and references therein. If the regularization term is quartic, (p-RS) reduces to the double-well potential optimization, which has particular applications in solid mechanics and quantum mechanics [21,22].…”

Section: Introductionmentioning

confidence: 99%

Trust-region and $p$-regularized subproblems: local nonglobal minimum is the second smallest objective function value among all first-order stationary points

Wang,

Song,

Xia

2021

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

Cited by 5 publications

References 56 publications

On the p-regularized trust region subproblem

On the p-regularized trust region subproblem

S-lemma with equality and its applications

Trust-region and $p$-regularized subproblems: local nonglobal minimum is the second smallest objective function value among all first-order stationary points

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