---
title: "optimization-solvers — Mova code model"
description: "Mova code model of models/optimization-solvers.groupant: components with their labels, obernants and the model text."
url: "https://groupants.elamur.ai/free/en/mova/models/optimization-solvers.groupant/"
---
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# optimization-solvers

The groupant beside models/optimization-solvers.groupant in the Mova repository: what the code computes, what it never does, how to go back from its outputs, its invariants, cost and checks.

## Facts

- **kind:** groupant_group
- **status:** partial source bound group
- **bound sources:** 8
- **repository folder:** [`models/optimization-solvers.groupant`](https://github.com/ElamurAI/mova-free/tree/main/models/optimization-solvers.groupant)
- **repository commit:** `855da7bd4a7b`

> **Implementation material.** A Mova code model describes this code at one source revision: partial models, finite checks and their stated scope. It is not a reviewed general exposition, and its labels are shown as written, never raised.

## Components

| component | label | where |
|---|---|---|
| **approximant** | partial source model | [forward](#forward) |
| **ast** | not materialized |  |
| **composition** | partial source model | [composition](#composition) |
| **cost** | partial source model | [cost](#cost) |
| **descriptant** | partial source model | [epoch-memory](#epoch-memory) |
| **invariant** | partial source model | [invariant](#invariant) |
| **tests** | bounded executed | [checks](#checks) |

## Obernants

The typed relations around the forward model: the behaviour complement (what the code never does), the reverse reconstruction (every input compatible with an output) and specialised views.

| id | type | label | where |
|---|---|---|---|
| `optimization-solvers-behavior-complement` | behavior_complement | partial source model | [behavior-complement](#behavior-complement) |
| `optimization-solvers-reverse-reconstruction` | reverse_reconstruction | partial source model | [reverse-reconstruction](#reverse-reconstruction) |
| `optimization-solvers-scalar-stationarity` | stationarity_transport | partial source model | [scalar-stationarity](#scalar-stationarity) |
| `optimization-solvers-diagonal-equations` | residual_transport | partial source model | [diagonal-equations](#diagonal-equations) |
| `optimization-solvers-conditioning` | conditional_error_bound | partial source model | [conditioning](#conditioning) |

## Model text


### model

## optimization-solvers

### forward

Bracket roots and deterministic strictly unimodal minima; descend diagonal quadratics and solve dense symmetric positive definite linear systems with explicit stopping observations.

### invariant

Preserve finite reports, bracket order, dimensional agreement and distinct converged, exhausted and stagnated states. True residual is recomputed for stopping.

### cost

Sum O(k_b+k_g+n*k_d+n^2*k_c) arithmetic plus callback costs; serial auxiliary O(n), input matrix O(n^2). Hard dimension and iteration limits remain leaf-specific.

### composition

On q(x)=d*x^2/2-b*x with d>0, derivative root, scalar minimizer, diagonal descent and one-dimensional CG share b/d. For a common diagonal system, GD and CG target identical stationarity but need not share iteration counts or statuses.

### epoch-memory

Wave19, 2026-10-30. Models retain real-arithmetic assumptions separately from native finite checks. Kernels are library tools, not automatically adopted into training or inference.

### checks

["quadratic_solvers_join_scalar_roots_and_polynomial_values"]

### limitations

Continuity, deterministic strict unimodality and global SPD are not established by sampling or visited curvature. Residual magnitude alone does not bound coordinate error without conditioning. Floating-point underflow and conservative arithmetic rejection remain possible.

### catalog

## Typed family relations

Each relation keeps its own domain and assurance.

### behavior-complement

Type: behavior_complement.

Exclude input/report pairs outside each tagged solver relation: exhausted or stagnant reports relabeled as convergence, bracket-width bounds promoted to objective bounds, residual norms promoted to unconditional coordinate error, or visited positive curvature promoted to global SPD.

### reverse-reconstruction

Type: reverse_reconstruction.

A returned estimate and status admit many callbacks, brackets, initial states and budgets. For zero residual x, any positive diagonal D with b=D*x is compatible. Common roots/minima survive positive scaling; output does not uniquely recover the problem.

### scalar-stationarity

Type: stationarity_transport.

For q(x)=d*x^2/2-b*x,d>0 with valid enclosing intervals, q'=d*x-b links bisection, golden-section, GD and CG to the same b/d. Native finite fixtures compare additive coordinates with explicit tolerance, not status equality.

### diagonal-equations

Type: residual_transport.

For the same positive diagonal D and b, gradient D*x-b and linear residual b-D*x differ only by sign. Their Euclidean norms agree in real arithmetic; rounding and solver trajectories remain separate.

### conditioning

Type: conditional_error_bound.

For SPD A with known lambda_min>0, ||x-x*||2 <= ||A*x-b||2/lambda_min in real arithmetic. This group supplies no eigenvalue certificate; without that premise the residual-to-solution bound is absent.

### wave19-evidence

#### Family evidence — 2026-10-30

This family binds eight implementation/test sources through its manifest; views are partial source-derived models. Selected named joins are ["quadratic_solvers_join_scalar_roots_and_polynomial_values"] in algorithms/tests/composition_wave19.rs. Those checks pass four times as part of the 616-check suite. Leaf oracles and compiled production controls remain separately bound in adjacent evidence.

[Integrated evidence](https://md.groupants.elamur.ai/free/en/mova/models/algorithm-library.groupant/#wave19-evidence) records receipts; conditional laws outside these fixture scopes are not promoted to executed checks. No joint universal percentage bound follows from family membership.