Full-Matrix Gauss-Newton Optimization

• Full-matrix Gauss-Newton optimization is a method that computes the complete second-order curvature matrix to enhance convergence in nonlinear least-squares and composite convex problems.
• It employs a majorant condition and quasi-regularity to establish semi-local convergence guarantees, extending classical assumptions with practical error bounds.
• Practical implementations leverage adaptive step sizing and error control to robustly solve high-dimensional, complex optimization tasks in scientific computing and machine learning.

Full-matrix Gauss–Newton optimization refers to the use of the classical Gauss–Newton algorithm—specifically, the variant that computes and employs the full (dense) second-order curvature matrix arising from the quadratic linearization of nonlinear least-squares or composite convex problems. This approach is foundational to nonlinear optimization theory and has been extended to convex composite objectives, large-scale scientific computing, and high-dimensional machine learning. Recent advances emphasize convergence guarantees under mild conditions, extend the applicability beyond locally well-behaved cases, and elucidate implementation details for practical full-matrix computations.

1. Majorant Condition and Relaxed Regularity

A central innovation in modern convergence theory for full-matrix Gauss–Newton optimization is the “majorant condition,” which generalizes (and often weakens) classical assumptions about the regularity of the nonlinear operator. Instead of requiring a global Lipschitz condition on the Jacobian $F'(x)$, the analysis is organized around a scalar, twice-differentiable majorant function $f : [0, R) \to \mathbb{R}$ that controls the variation of $F'(x)$ within a ball $B(x_0, R) \subset \mathbb{R}^n$:

$$\|F'(y) - F'(x)\| \leq f'(\|y - x\| + \|x - x_0\|) - f'(\|x - x_0\|), \qquad \forall x, y \in B(x_0, R),\,\|x - x_0\| + \|y - x\| < R$$

with $f(0) = 0$, $f'(0) = -1$, and $f'$ convex and strictly increasing.

This approach allows one to “majorize” the nonlinear problem by comparing it to a one-dimensional Newton iteration applied to an auxiliary function $f_{\sigma, a}(t) = \sigma + (a-1)t + a f(t)$, with convergence regions defined by the zero $t^*$ of $f_{\sigma, a}$. The contraction properties and well-behavedness of the nonlinear iteration are inferred from the properties of the auxiliary scalar sequence, rendering the overall proof more transparent than classical Lipschitz arguments (Ferreira et al., 2011).

2. Quasi-Regular Points and Error Propagation

The convergence results assume that the initial iterate $x_0$ is a quasi-regular point for the inclusion $F(x) \in C$, where $$C = \{z \in \mathbb{R}^m: h(z) \leq h(x) \text{ for some } x \in \mathbb{R}^n\}$$ and $h$ is convex. Quasi-regularity requires that for all $x$ in a neighborhood $B(x_0, r)$, the distance to the linearized solution set $D_C(x)$ satisfies:

$d(0, D_C(x)) \leq \beta(\|x - x_0\|)\, d(F(x), C)$

for some nondecreasing, positive function $\beta$. Here, $D_C(x)$ is the set of directions $d$ for which the linearization $F(x) + F'(x)d \in C$. This generalizes classical regularity and ensures that the search directions obtained via the local convex subproblems are robust, so that the nonlinear and majorant errors remain tightly linked (Ferreira et al., 2011).

3. Semi-Local Convergence and Regions of Good Behavior

The analysis delivers semi-local convergence guarantees—properties are valid within an explicit neighborhood of the initial point, determined by the majorant function and quasi-regularity, rather than only locally near a precise solution or globally on $\mathbb{R}^n$. Typically, the region of “well-behavedness” is a ball around $x_0$ with radius related to $t^*$ (the zero of $f_{\sigma,a}$), and the Gauss–Newton sequence remains inside this region as long as the model assumptions are satisfied. This semi-local character is more practical than purely local results: the algorithm can be initiated at greater distances from a solution than classical Kantorovich-type analyses require, and convergence is still quantified (Ferreira et al., 2011).

4. Application to Convex Composite Optimization

Full-matrix Gauss–Newton optimization is not limited to unconstrained or purely least-squares problems. For composite convex objectives of the form

$\min_{x \in \mathbb{R}^n} h(F(x))$

with $h: \mathbb{R}^m \to \mathbb{R}$ convex and $F: \mathbb{R}^n \to \mathbb{R}^m$ continuously differentiable, the method involves, at each iteration, the solution of a convex subproblem:

$$d_k \in \operatorname*{arg\,min}_{d \in \mathbb{R}^n,\,\|d\| \leq A} h(F(x_k) + F'(x_k) d)$$

followed by the update $x_{k+1} = x_k + d_k$. This extension is critical for a broad range of applications including penalization, minimax, and goal programming, and the majorant-based theory supports the convergence of such algorithms beyond standard nonconvex or smooth settings (Ferreira et al., 2011).

5. Mathematical Structures and Error Bounds

The backbone of the analysis consists of explicit inequalities and auxiliary sequences:

• Majorant function inequality:

$$\|F'(y) - F'(x)\| \leq f'(\|y - x\| + \|x - x_0\|) - f'(\|x - x_0\|)$$

• Newton iteration for auxiliary function:

$$t_{k+1} = t_k - \frac{f_{\sigma,a}(t_k)}{f'_{\sigma,a}(t_k)}$$

• Error control:

$\|x_{k+1} - x_k\| \leq t_{k+1} - t_k$

• Quasi-regularity:

$d(0, D_C(x)) \leq \beta(\|x - x_0\|)\, d(F(x), C)$

These formulas allow reductions from the nonlinear multivariate problem to an auxiliary scalar model, so that analysis of convergence, including contraction and step restriction, parallels that of Newton’s method in the majorant space (Ferreira et al., 2011).

6. Practical Guidelines and Implementation Insights

The semi-local majorant-based analysis provides multiple practical consequences for implementing full-matrix Gauss–Newton optimization:

• Verification of assumptions in applications is reduced to checking properties of a scalar function, rather than multidimensional regularity.
• The allowable step lengths and progress per iteration are quantitatively related to the auxiliary Newton sequence, providing direct implementable bounds for stopping criteria and adaptive step size selection.
• The approach admits scenarios where $F'$ is not globally Lipschitz, as long as the majorant condition holds, and allows for initial points outside sets “classically” considered regular.
• If further conditions (such as strict convexity for $f'_{\sigma,a}$) are met, Q-quadratic convergence rates follow, enhancing the speed of convergence in practice.

This framework is hence relevant for developing robust, efficient solvers for convex composite and nonlinear optimization without the need for restrictive regularity conditions, and forms the foundation for modern convergence analyses in full-matrix Gauss–Newton methods (Ferreira et al., 2011).

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In summary, the convergence of full-matrix Gauss–Newton optimization under majorant conditions and quasi-regularity unifies and extends classical theory, supports robust algorithmic implementation, and provides explicit quantitative convergence rates across a wide class of convex composite optimization problems, highlighted by the interplay between scalar auxiliary majorant models and the underlying nonlinear system.