---
title: "algorithm-projected-gradient-box — Mova code model"
description: "Mova code model of algorithms/src/projected_gradient_box/mod.rs: components with their labels, obernants and the model text."
url: "https://groupants.elamur.ai/free/en/mova/algorithms/src/projected_gradient_box/mod.rs.groupant/"
---
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# algorithm-projected-gradient-box

The groupant beside algorithms/src/projected_gradient_box/mod.rs 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
- **status:** partial descriptive instance
- **source:** [`algorithms/src/projected_gradient_box/mod.rs`](https://github.com/ElamurAI/mova-free/blob/main/algorithms/src/projected_gradient_box/mod.rs)
- **source SHA-256:** `53e3dcbf5541e1ef…`
- **repository folder:** [`algorithms/src/projected_gradient_box/mod.rs.groupant`](https://github.com/ElamurAI/mova-free/tree/main/algorithms/src/projected_gradient_box/mod.rs.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 | [approximant](#approximant) |
| **ast** | partial | `syntax.json` |
| **descriptant** | present | [descriptant](#descriptant) |
| **invariant** | partial | [approximant](#approximant) |
| **tests** | finite checks | [tests](#tests) |

## 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 |
|---|---|---|---|
| `behavior_complement` | behavior_complement | partial relational specification | [behavior_complement](#behavior-complement) |
| `reverse_reconstruction` | reverse_reconstruction | partial relational specification | [reverse_reconstruction](#reverse-reconstruction) |
| `update_relation` | update_relation | partial relational specification | [update_relation](#update-relation) |
| `termination_partition` | termination_partition | partial relational specification | [termination_partition](#termination-partition) |
| `finite_arithmetic` | finite_arithmetic | partial relational specification | [finite_arithmetic](#finite-arithmetic) |
| `resource_observation` | resource_observation | partial relational specification | [resource_observation](#resource-observation) |

## Model text


### descriptant

#### projected_gradient_box implementation

Minimize f(x)=.5*x'Q*x-b'x on lower<=x<=upper. Validate exact symmetry and nonnegative diagonal; off-diagonal row absolute sum must be zero or below diagonal/(1+128*EPSILON), a conservative sufficient PSD domain. L=max(max row absolute sum,1e-100). Clip initial point, then x_next=clip(x-(Qx-b)/L). Residual=max_i|x_i-x_next_i| is a projected-step observation, not gradient norm or solution distance.

##### Prospective epoch

The prospective model was saved before source creation in the external wave20 optimization run and retained in prospective.md. Original standard-library Rust, no downloaded solution code.

##### Observations retained

13 diagonal cases compared to clipped analytic minimizers; dense 2x2 stationary optimizer [1,2] and objective -7.5; linear objective with zero Q; zero budget; negative diagonal, asymmetric and empty matrix rejection.

##### Limits

Projected-step residual<=tolerance precedes budget exhaustion. Each update swaps buffers, preserving coordinate order. The equality stagnation guard is presently dominated by the residual-zero convergence test for finite values and positive tolerance; Status::Stagnated is shared but this branch is unreachable in this scoped arithmetic model. No unsupported stagnation coverage is claimed. No timing or universal machine-accuracy claim.

Reference: Bertsekas, Nonlinear Programming (1999); Nocedal and Wright, Numerical Optimization (2006).

### model

#### Mathematical combiner

Inputs are the authoritative mod.rs, source-bound partial syntax, forward relation, typed complement first and reverse reconstruction second, specialized relations and named finite evidence. Intersect the accepted-domain relation with each update relation, then partition termination by convergence, budget, stagnation or error. Minimize f(x)=.5*x'Q*x-b'x on lower<=x<=upper. Validate exact symmetry and nonnegative diagonal; off-diagonal row absolute sum must be zero or below diagonal/(1+128*EPSILON), a conservative sufficient PSD domain. L=max(max row absolute sum,1e-100). Clip initial point, then x_next=clip(x-(Qx-b)/L). Residual=max_i|x_i-x_next_i| is a projected-step observation, not gradient norm or solution distance.

Tests are sibling source nodes; manifest children are absent because these compact files have no independent nested fragment contracts. Syntax does not expand macros or prove semantics.

Retained observation: 13 diagonal cases compared to clipped analytic minimizers; dense 2x2 stationary optimizer [1,2] and objective -7.5; linear objective with zero Q; zero budget; negative diagonal, asymmetric and empty matrix rejection.

Composition limits: Dimension/value/matrix validation precedes bound validation, both precede checked allocation of x, gradient, next. Failed try_reserve maps to AllocationFailed. Iteration order ascending, no callbacks. Caller matrix memory O(n²) excluded from scratch cost. Zero Q uses tiny positive scale and often jumps to an active box endpoint. Projected-step residual<=tolerance precedes budget exhaustion. Each update swaps buffers, preserving coordinate order. The equality stagnation guard is presently dominated by the residual-zero convergence test for finite values and positive tolerance; Status::Stagnated is shared but this branch is unreachable in this scoped arithmetic model. No unsupported stagnation coverage is claimed.

### approximant

#### Forward approximant and invariant

Minimize f(x)=.5*x'Q*x-b'x on lower<=x<=upper. Validate exact symmetry and nonnegative diagonal; off-diagonal row absolute sum must be zero or below diagonal/(1+128*EPSILON), a conservative sufficient PSD domain. L=max(max row absolute sum,1e-100). Clip initial point, then x_next=clip(x-(Qx-b)/L). Residual=max_i|x_i-x_next_i| is a projected-step observation, not gradient norm or solution distance.

##### Domain and invariant

n in 1..64, finite operands abs<=1e50, exact rectangular shape and symmetry, finite positive tolerance, budget<=100000; then finite ordered bounds. Nonfinite objective, gradient or raw step is NonFiniteArithmetic. Summed diagonal dominance includes a safety margin but is not a general floating PSD certification.

Coordinate clipping is separable and idempotent. For real arithmetic with L a Lipschitz gradient bound, the projection variational inequality implies nonincreasing objective. Floating code checks finite objective/gradient/raw steps, not a machine-level descent proof. Shared validation/evaluate/buffer helpers are consumed by Frank-Wolfe; their domain and error mapping are part of the wiring.

##### Cost

O(n²) validation and O(k*n²) arithmetic; O(n) additional storage (three length-n vectors), n<=64. Dimension/value/matrix validation precedes bound validation, both precede checked allocation of x, gradient, next. Failed try_reserve maps to AllocationFailed. Iteration order ascending, no callbacks. Caller matrix memory O(n²) excluded from scratch cost. Zero Q uses tiny positive scale and often jumps to an active box endpoint.

##### Executable approximation gap

No useful percentage-bounded cheaper surrogate is attached. A sampled decrease, projected step, bracket width or KKT/dual residual is not a universal relative objective/solution certificate. No 1/2/5/10% claim is inferred from finite success; zero-reference relative error is undefined without a separately specified metric. This gap is explicit rather than wrapping the exact implementation.

Reference: Bertsekas, Nonlinear Programming (1999); Nocedal and Wright, Numerical Optimization (2006).

### models

## Typed obernant family

These are scoped partial relations, not executable inverses or semantic proofs.

### behavior_complement

Accepted iterates cannot lie outside their finite box. Converged does not imply unconstrained gradient zero: an active upper-bound coordinate can have negative gradient. General PSD matrices outside the conservative diagonal-dominance domain are intentionally rejected; no universal relative-error bound follows from the step residual.

### reverse_reconstruction

For interior coordinates in exact arithmetic, an observed step fixes (Qx-b)_i=L*(x_i-x_next_i). A clipped lower endpoint only constrains the raw step to <=lower; an upper endpoint only constrains it to >=upper. Thus many gradients and original initial values project to the same output. A zero residual constrains KKT signs rather than uniquely recovering b or Q.

### update_relation

Coordinate clipping is separable and idempotent. For real arithmetic with L a Lipschitz gradient bound, the projection variational inequality implies nonincreasing objective. Floating code checks finite objective/gradient/raw steps, not a machine-level descent proof. Shared validation/evaluate/buffer helpers are consumed by Frank-Wolfe; their domain and error mapping are part of the wiring.

### termination_partition

Projected-step residual<=tolerance precedes budget exhaustion. Each update swaps buffers, preserving coordinate order. The equality stagnation guard is presently dominated by the residual-zero convergence test for finite values and positive tolerance; Status::Stagnated is shared but this branch is unreachable in this scoped arithmetic model. No unsupported stagnation coverage is claimed.

### finite_arithmetic

n in 1..64, finite operands abs<=1e50, exact rectangular shape and symmetry, finite positive tolerance, budget<=100000; then finite ordered bounds. Nonfinite objective, gradient or raw step is NonFiniteArithmetic. Summed diagonal dominance includes a safety margin but is not a general floating PSD certification.

### resource_observation

Dimension/value/matrix validation precedes bound validation, both precede checked allocation of x, gradient, next. Failed try_reserve maps to AllocationFailed. Iteration order ascending, no callbacks. Caller matrix memory O(n²) excluded from scratch cost. Zero Q uses tiny positive scale and often jumps to an active box endpoint.


### tests

#### Executable checks

Run `cargo test -p mova-algorithms projected_gradient_box::tests`.

13 diagonal cases compared to clipped analytic minimizers; dense 2x2 stationary optimizer [1,2] and objective -7.5; linear objective with zero Q; zero budget; negative diagonal, asymmetric and empty matrix rejection.

Independent oracle: Diagonal Q has exact optimizer clip(b_i/Q_ii); dense example directly satisfies Q*[1,2]=[5,5]. These independent identities verify location and objective without replaying projected iterations.

Compiled negative control: Change shared gradient dot-b to dot+b; diagonal_clipping_oracle must reject the compiled incorrect optimization direction.

No allocator failure injection, all-input proof or benchmark is claimed.

### prospective

#### Prospective optimization model
Before source creation. Brent maintains unimodal bracket and three sampled points; accepts safeguarded parabolic proposals, otherwise golden contraction. O(k) callbacks, O(1) storage. Deterministic callback is assumed; strict unimodality cannot be checked. Armijo shrinks positive alpha by rho until f(alpha)<=f0+c*alpha*slope, slope<0. O(k) callbacks, O(1) space; supplied f0 and slope are caller premises.
Dense quadratics f=.5*x'Qx-b'x use symmetric diagonally dominant nonnegative Q, dimension<=64, finite magnitudes<=1e50. This is a sufficient restricted PSD domain, not arbitrary convex quadratics. PG clips x-g/L into finite bounds with L=max absolute row sum; mapping residual is max |x-clip(x-g/L)|. FW on unit simplex selects earliest minimum gradient vertex, gap=g'(x-s), step=min(1,gap/(d'Qd)); linear curvature uses step 1. O(kn²) time O(n) scratch. These real identities are not floating error guarantees.
Lasso minimizes .5||Ax-y||²+lambda||x||1 by cyclic exact coordinate minimization and residual maintenance, using existing soft threshold kernel. Up to 256 rows and 64 columns. O(kmn) time O(m+n) space. Stop on recomputed KKT max violation. Zero columns set zero. Deterministic ties and iteration orders. All validation precedes work; checked allocation and nonfinite arithmetic explicit; budget<=100000, tolerance positive finite. Each routine distinguishes convergence, budget exhaustion and representational stagnation. No percent objective/solution bound. Expected algorithmic advantage: interpolation avoids golden-only work on smooth functions; cached residual updates avoid recomputing predictions per coordinate. No measured speed claim. Executable approximate surrogate deferred: residual and finite samples lack a justified universal supported percentage metric.
References: Brent, Algorithms for Minimization without Derivatives (1973); Armijo (1966); Nocedal and Wright (2006); Friedman et al., Regularization Paths for Generalized Linear Models via Coordinate Descent (2010); Frank and Wolfe (1956).

### validation

#### Validation epoch 2026-10-30

External receipts: `$MOVA_DATA/runs/octopus-wave20-optimization-2026-10-30`. Two debug and two release library runs each pass 568 tests, including 21 new native tests, with identical sorted test-name/outcome transcripts. Strict all-target algorithms Clippy and focused rustfmt pass.

Five independently compiled production mutants are rejected by named runtime assertions; source bytes restored before the positive rerun. This source's control: Change shared gradient dot-b to dot+b; diagonal_clipping_oracle must reject the compiled incorrect optimization direction.

Fresh syntax projections are generated twice and compared before installing syntax.json. These are partial structural observations, not macro-expanded AST or semantic proofs. No performance or universal <=10% result follows.