Convexity of the longitudinal variation of third-order resonance driving terms and its application in dynamic aperture optimization

Abstract: The optimization of the dynamic aperture (DA) of a storage ring is typically a non-convex problem with multiple local optima. Recent studies showed that reducing the variation of resonance driving terms (RDTs) along the longitudinal position improves DA very effectively, as the reduction in the longitudinal variation of lower-order RDTs suppresses higher-order nonlinear terms. Therefore, minimizing the longitudinal variation of third-order RDTs is crucial for DA optimization. In this paper, we prove that the longitudinal variation of third-order RDTs, quantified using their RMS value $f_{3,\mathrm{rms}}$ at sextupole locations, is a special convex function. In the space of sextupole strengths, the iso-surfaces of $f_{3,\mathrm{rms}}$ are a series of concentric and coaxial ellipsoidal surfaces, with the central position possessing minimum $f_{3,\mathrm{rms}}$. The scanning results of a storage ring lattice show a strong consistency between the distributions of $f_{3,\mathrm{rms}}$ and DA, indicating that the optimization of DA can be regarded as a roughly approximate convex optimization problem. Based on this, a fast DA optimization method based on particle tracking is developed, where a high-quality initial population for an intelligent algorithm is generated with a Gaussian distribution based on the geometric structure of $f_{3,\mathrm{rms}}$.

• The paper proves that the RMS longitudinal variation of third-order resonance driving terms is a positive-definite quadratic norm with concentric ellipsoidal contours in sextupole-strength space, including under chromaticity constraints.
• The paper shows in FODO and HLS-III studies that lower third-order RDT variation generally corresponds to larger dynamic aperture and lower frequency diffusion, while noting exceptions caused by amplitude-dependent tune-shift effects.
• The paper uses the fitted quadratic geometry to initialize differential-evolution searches with an aligned Gaussian population, achieving better SSRF optimization results after 10 generations than random initialization achieved after 50 generations, though the method remains limited to on-momentum, third-order optimization.

Background and motivation

Dynamic aperture (DA) is a decisive performance metric for storage rings, governing injection efficiency and beam lifetime. Its optimization over the sextupole strength space is a non-convex problem with multiple local optima, and the difficulty grows as natural emittance decreases and the number of sextupole families increases. Existing approaches divide into numerical methods—scanning and intelligent algorithms operating directly on particle tracking—which can in principle find the global optimum but are computationally expensive and offer little physical insight, and analytical methods, chiefly the minimization of resonance driving terms (RDTs) combined with frequency map analysis (FMA) and amplitude-dependent tune shift (ADTS) control.

Building on prior work showing that reducing the longitudinal variation of RDTs enlarges the DA far more effectively than reducing the conventional one-turn RDTs [Wei et al., PRAB 26, 084001 (2023)], this paper establishes a structural property of that variation: the RMS value of third-order RDTs at sextupole locations, $f_{3,\mathrm{rms}}$, is a special convex function whose iso-surfaces are concentric, coaxial ellipsoids in the sextupole strength space. This convexity is then exploited both to reinterpret the DA optimization landscape and to construct a faster tracking-based optimization scheme.

Physical mechanism and FODO demonstration

The argument rests on the Baker–Campbell–Hausdorff expansion of the one-turn Lie map. Writing the accumulated first- and second-order terms as $A_t^{(1)} = \sum_{i=1}^{t}\hat{V}_i$ and $$A_t^{(2)} = \frac{1}{2}\sum_{j=2}^{t}\{A_j^{(1)}, \hat{V}_j\}$$, the higher-order terms of $e^{:h:}$ are generated through the longitudinal accumulation of lower-order ones. Consequently, minimizing the variation of third-order RDTs along the ring suppresses higher-order RDTs and controls ADTSs, thereby enlarging the DA. The longitudinal variation is quantified as

$$f_{3,\mathrm{rms}} = \sqrt{\sum_{3=j+k+l+m} f_{jklm,\mathrm{rms}}^2}, \qquad f_{jklm,\mathrm{rms}} = \sqrt{\frac{1}{N}\sum_{i=1}^{N}|f_{jklm}(z_i)|^2},$$

with $z_i$ the sextupole locations and $f_{jklm}(z)$ the standard geometric RDT expression.

A FODO lattice with four thin-lens sextupole families, chromaticities corrected to $(1.0, 1.0)$, and 500 randomly generated nonlinear solutions provides the empirical demonstration. Across these solutions, the surviving particle count $N_{\text{particle}}$ increases monotonically with decreasing $f_{3,\mathrm{rms}}$, and for fixed $A_t^{(1)} = \sum_{i=1}^{t}\hat{V}_i$0 the average frequency diffusion rate $A_t^{(1)} = \sum_{i=1}^{t}\hat{V}_i$1 also decreases with $A_t^{(1)} = \sum_{i=1}^{t}\hat{V}_i$2—so reducing $A_t^{(1)} = \sum_{i=1}^{t}\hat{V}_i$3 improves both DA size and motion stability. The correlation with the one-turn RDT metric $A_t^{(1)} = \sum_{i=1}^{t}\hat{V}_i$4 is markedly weaker, with no clear relationship to $A_t^{(1)} = \sum_{i=1}^{t}\hat{V}_i$5. One caveat is acknowledged: a subset of solutions within a small $A_t^{(1)} = \sum_{i=1}^{t}\hat{V}_i$6 region deviates due to unusually small ADTSs, indicating that $A_t^{(1)} = \sum_{i=1}^{t}\hat{V}_i$7 is not the sole determinant of DA quality.

Proof of convexity

The central theoretical result is that $A_t^{(1)} = \sum_{i=1}^{t}\hat{V}_i$8 is a quadratic norm. Since the RDT coefficient $A_t^{(1)} = \sum_{i=1}^{t}\hat{V}_i$9 is linear in the normalized sextupole strength $$A_t^{(2)} = \frac{1}{2}\sum_{j=2}^{t}\{A_j^{(1)}, \hat{V}_j\}$$0, each $$A_t^{(2)} = \frac{1}{2}\sum_{j=2}^{t}\{A_j^{(1)}, \hat{V}_j\}$$1 expands as a real quadratic form in the strengths; summing over RDTs and sextupole locations yields

$$A_t^{(2)} = \frac{1}{2}\sum_{j=2}^{t}\{A_j^{(1)}, \hat{V}_j\}$$2

where $$A_t^{(2)} = \frac{1}{2}\sum_{j=2}^{t}\{A_j^{(1)}, \hat{V}_j\}$$3 is real symmetric by construction. Positivity of $$A_t^{(2)} = \frac{1}{2}\sum_{j=2}^{t}\{A_j^{(1)}, \hat{V}_j\}$$4 follows from the non-negativity of $$A_t^{(2)} = \frac{1}{2}\sum_{j=2}^{t}\{A_j^{(1)}, \hat{V}_j\}$$5 for all nonzero $$A_t^{(2)} = \frac{1}{2}\sum_{j=2}^{t}\{A_j^{(1)}, \hat{V}_j\}$$6, so $$A_t^{(2)} = \frac{1}{2}\sum_{j=2}^{t}\{A_j^{(1)}, \hat{V}_j\}$$7 is positive definite and $$A_t^{(2)} = \frac{1}{2}\sum_{j=2}^{t}\{A_j^{(1)}, \hat{V}_j\}$$8 is a quadratic norm—convex on $$A_t^{(2)} = \frac{1}{2}\sum_{j=2}^{t}\{A_j^{(1)}, \hat{V}_j\}$$9, with ellipsoidal unit balls. Under the two linear chromaticity correction constraints, the feasible set is an $e^{:h:}$0-dimensional affine subspace (hence convex), on which $e^{:h:}$1 remains convex; after eliminating the constraints it takes the form

$e^{:h:}$2

whose minimum—the ellipsoid center—can be obtained analytically once $e^{:h:}$3, $e^{:h:}$4, and $e^{:h:}$5 are fitted. Scans of the FODO sextupole family strengths confirm elliptical contours in both the constrained and unconstrained cases, with the constrained ellipsoid center shifted from the origin. The paper notes that $e^{:h:}$6, $e^{:h:}$7, and $e^{:h:}$8 share the same mathematical form and are therefore also convex; however, the analogous fourth-order RDT variation is explicitly found not to be convex, so the result does not generalize to fourth order.

Consistency with the DA landscape: HLS-III scans

Global scans of the HLS-III storage ring lattice, with chromaticities fixed at $e^{:h:}$9 by straight-section sextupoles and two (then three) arc-section families as free variables, compare the distributions of $$f_{3,\mathrm{rms}} = \sqrt{\sum_{3=j+k+l+m} f_{jklm,\mathrm{rms}}^2}, \qquad f_{jklm,\mathrm{rms}} = \sqrt{\frac{1}{N}\sum_{i=1}^{N}|f_{jklm}(z_i)|^2},$$0 and the total frequency diffusion rate $$f_{3,\mathrm{rms}} = \sqrt{\sum_{3=j+k+l+m} f_{jklm,\mathrm{rms}}^2}, \qquad f_{jklm,\mathrm{rms}} = \sqrt{\frac{1}{N}\sum_{i=1}^{N}|f_{jklm}(z_i)|^2},$$1 of surviving particles. In both the four-family and five-family cases, the contours of $$f_{3,\mathrm{rms}} = \sqrt{\sum_{3=j+k+l+m} f_{jklm,\mathrm{rms}}^2}, \qquad f_{jklm,\mathrm{rms}} = \sqrt{\frac{1}{N}\sum_{i=1}^{N}|f_{jklm}(z_i)|^2},$$2 are concentric coaxial ellipses, and the distributions of $$f_{3,\mathrm{rms}} = \sqrt{\sum_{3=j+k+l+m} f_{jklm,\mathrm{rms}}^2}, \qquad f_{jklm,\mathrm{rms}} = \sqrt{\frac{1}{N}\sum_{i=1}^{N}|f_{jklm}(z_i)|^2},$$3 and $$f_{3,\mathrm{rms}} = \sqrt{\sum_{3=j+k+l+m} f_{jklm,\mathrm{rms}}^2}, \qquad f_{jklm,\mathrm{rms}} = \sqrt{\frac{1}{N}\sum_{i=1}^{N}|f_{jklm}(z_i)|^2},$$4 exhibit strong consistency: low-$$f_{3,\mathrm{rms}} = \sqrt{\sum_{3=j+k+l+m} f_{jklm,\mathrm{rms}}^2}, \qquad f_{jklm,\mathrm{rms}} = \sqrt{\frac{1}{N}\sum_{i=1}^{N}|f_{jklm}(z_i)|^2},$$5 regions coincide broadly with low-$$f_{3,\mathrm{rms}} = \sqrt{\sum_{3=j+k+l+m} f_{jklm,\mathrm{rms}}^2}, \qquad f_{jklm,\mathrm{rms}} = \sqrt{\frac{1}{N}\sum_{i=1}^{N}|f_{jklm}(z_i)|^2},$$6 regions. Notably, in the five-family case the two extreme regions of $$f_{3,\mathrm{rms}} = \sqrt{\sum_{3=j+k+l+m} f_{jklm,\mathrm{rms}}^2}, \qquad f_{jklm,\mathrm{rms}} = \sqrt{\frac{1}{N}\sum_{i=1}^{N}|f_{jklm}(z_i)|^2},$$7 are not discrete islands but lie on the same connected region inside the central zone of the $$f_{3,\mathrm{rms}} = \sqrt{\sum_{3=j+k+l+m} f_{jklm,\mathrm{rms}}^2}, \qquad f_{jklm,\mathrm{rms}} = \sqrt{\frac{1}{N}\sum_{i=1}^{N}|f_{jklm}(z_i)|^2},$$8 ellipsoid—contradicting the common expectation that DA exhibits multiple isolated optima. Two selected solutions with quite different strengths (e.g., S5 of $$f_{3,\mathrm{rms}} = \sqrt{\sum_{3=j+k+l+m} f_{jklm,\mathrm{rms}}^2}, \qquad f_{jklm,\mathrm{rms}} = \sqrt{\frac{1}{N}\sum_{i=1}^{N}|f_{jklm}(z_i)|^2},$$9 vs. $z_i$0) have nearly identical $z_i$1 (16.52 vs. 16.66 $z_i$2) and comparable $z_i$3 ($z_i$4 vs. $z_i$5), and the authors cite the recent NSLS-II chaos-suppression result as further corroboration that distinct good solutions are connected through ellipsoidal surfaces of $z_i$6.

The inference that good DA solutions populate the ellipsoid center in higher-dimensional spaces is supported but not proven: it is an extrapolation from two- and three-family scans, and the paper frames the DA optimization as only "roughly approximate" convex on this basis.

Fast DA optimization on the SSRF lattice

Because $z_i$7 is a standard quadratic form, it matches the exponent structure of a multivariate Gaussian density. The proposed method fits $z_i$8, $z_i$9, $f_{jklm}(z)$0 from randomly generated solutions, locates the ellipsoid center, and draws the initial population from a Gaussian aligned with the ellipsoid, with $f_{jklm}(z)$1 set to $f_{jklm}(z)$2 times the minimum $f_{jklm}(z)$3 ($f_{jklm}(z)$4). On the SSRF lattice with eight sextupole families (six straight-section families as variables, chromaticities at $f_{jklm}(z)$5), differential evolution with population 50 over 50 generations, minimizing $f_{jklm}(z)$6 over three independent runs per scheme, shows that the Gaussian-initialized runs converge substantially faster: their results at the 10th generation outperform the random-initialization results at the 50th generation. The gain is attributed to the higher quality of the initial population and, indirectly, to the consistency between the $f_{jklm}(z)$7 and DA distributions in the eight-dimensional variable space.

Limitations and open questions

The paper is explicit about several boundaries of its claims. The analysis is restricted to on-momentum DA; off-momentum behavior is deferred to future work. The convexity result holds for third-order RDTs only, as the fourth-order analogue is demonstrably non-convex. The equivalence between the $f_{jklm}(z)$8 distribution and the DA distribution is empirical and approximate—exceptions arise when ADTSs dominate, as in the FODO subset—and the extension of the "good solutions near the ellipsoid center" picture to higher dimensions is inferred rather than established by exhaustive scanning. The method also assumes the fitted quadratic model of $f_{jklm}(z)$9 accurately represents the landscape, and its dependence on the grouping of sextupoles into families, while claimed to be benign, is not systematically explored. Whether the Gaussian-initialized scheme retains its advantage for multi-objective formulations including off-momentum DA and simultaneous linear optics optimization remains an open question the authors themselves identify.

Conclusion

The paper proves that the longitudinal variation of third-order RDTs, measured as $(1.0, 1.0)$0 at sextupole locations, is a quadratic norm—hence convex with concentric, coaxial ellipsoidal iso-surfaces in sextupole strength space, including under chromaticity constraints. Lattice scans (FODO, HLS-III) show strong consistency between this convex quantity and DA quality, suggesting that good solutions cluster in the ellipsoid's central region rather than in isolated islands. Exploiting this geometry via a Gaussian-distributed initial population markedly accelerates differential-evolution-based DA optimization on the SSRF lattice, with 10-generation Gaussian results exceeding 50-generation random results. The result connects a practically important non-convex accelerator optimization problem to a convex surrogate with explicit analytic structure, while leaving the off-momentum case, fourth-order terms, and the higher-dimensional DA–RDT correspondence as clearly stated open issues.