用于高阶信赖域和球面多项式优化的齐次张量框架
A Homogeneous Tensor Framework for High-Order Trust-Region and Spherical Polynomial Optimization
AI总结：
研究高阶方法中\(p\geq3\)阶泰勒子问题难求解的问题，基于齐次张量表示开发半径控制边界方法，聚焦三次情况引入PAM方法，嵌入Ada--HTM，在特定条件下达到自适应正则化类型评估复杂度，数值表现良好。
AI中文摘要：
高阶方法可改善最坏情况评估复杂度，但对于\(p\geq3\)阶，其泰勒子问题是非凸多项式优化问题，通常难以求解。我们基于齐次张量表示开发了一种半径控制边界方法。通过用常数坐标扩充步长，任何\(p\)阶泰勒多项式都可精确表示为\(p\)阶齐次张量形式；在规定半径下，边界模型是球面多项式优化问题。算法开发聚焦于\(p = 3\)的三次情况。对于球面上的非齐次三次式，引入二次平移并证明在明确平移界下与三模块多线性公式在全局最优时等价，这激发了具有闭式块更新的近端交替最小化（PAM）方法。将边界步长机制嵌入自适应齐次张量方法（Ada--HTM）。在明确的光滑性、保障下降、弱曲率非退化和局部细化条件下，Ada--HTM对于一阶平稳性达到自适应正则化类型（AR\(p\)型）评估复杂度\(\mathcal{O}(\epsilon^{-(p + 1)/p})\)。数值上，PAM在可处理认证的结构化三次实例上与二阶矩 - 平方和（SOS）证书匹配，对低秩张量扩展性特别好，使Ada--HTM与信赖域和三次正则化方法具有竞争力，在病态和严重缩放问题上收益最大。
英文摘要：
High-order methods can improve worst-case evaluation complexity, but for orders $p\geq3$ their Taylor subproblems are nonconvex polynomial optimization problems and are generally difficult to solve. We develop a radius-controlled boundary approach based on homogeneous tensor representations. By augmenting the step with a constant coordinate, any $p$th-order Taylor polynomial can be represented exactly as an order-$p$ homogeneous tensor form; at a prescribed radius, the boundary model is a spherical polynomial optimization problem. The representation applies to arbitrary $p$, while the algorithmic development focuses on the cubic case $p=3$. For an inhomogeneous cubic on the sphere, we introduce a quadratic shift and prove, under an explicit shift bound, equivalence with a three-block multilinear formulation at global optimality. This motivates a proximal alternating minimization (PAM) method with closed-form block updates; its objective values decrease and every accumulation point is stationary. We embed the boundary-step mechanism in an Adaptive Homogeneous Tensor Method (Ada--HTM). Under explicit smoothness, safeguarded-decrease, weak-curvature nondegeneracy, and local-refinement conditions, Ada--HTM attains the adaptive-regularization-type (AR$p$-type) evaluation complexity $\mathcal{O}(ε^{-(p+1)/p})$ for first-order stationarity. Numerically, PAM matches order-$2$ moment--sum-of-squares (SOS) certificates on the structured cubic instances for which certification is tractable, scales particularly well for low-rank tensors, and makes Ada--HTM competitive with trust-region and cubic-regularization methods, with its largest gains on ill-conditioned and badly-scaled problems.
