minimize · 1 variable · 1 constraint

Original Problem

minimize⁡∥x∥2\underset{}{\operatorname{minimize}} \quad \lVert {\mathbf{x}} \rVert_2
subject to\text{subject to}
1≤x1 \leq \mathbf{x}
✓Problem is DCP compliant.
1 variable, 1 constraint
t1=φ∥⋅∥2(x)t_{1} = \varphi^{\lVert\cdot\rVert_2}({\mathbf{x}})
convex⋅R+⋅1×1\text{convex} \cdot \mathbb{R}_+ \cdot 1×1
x\mathbf{x}
affine⋅?⋅3×1\text{affine} \cdot ? \cdot 3×1
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Constraints

[1] Inequality 3×1
11[constant, 1×1]
x\mathbf{x}[affine, 3×1]

Smith Form

minimize⁡t1\operatorname{minimize} \quad t_{1}
subject to:
(1)
t1=φ∥⋅∥2(x)t_{1} = \varphi^{\lVert\cdot\rVert_2}({\mathbf{x}})
constraints:
[1]
Inequality3×1\text{Inequality} \quad 3\times 1

Relaxed Smith Form

minimize⁡t1\operatorname{minimize} \quad t_{1}
subject to:
(1)
t1≥φ∥⋅∥2(x)t_{1} \geq \varphi^{\lVert\cdot\rVert_2}({\mathbf{x}})
constraints:
[1]
Inequality3×1\text{Inequality} \quad 3\times 1

Conic Form

minimize⁡t1\operatorname{minimize} \quad t_{1}
subject to:
(1)
(t1,x)∈Qn+1(t_{1}, {\mathbf{x}}) \in \mathcal{Q}^{n+1}
constraints:
[1]
Inequality3×1\text{Inequality} \quad 3\times 1

Standard Cone Form

min⁡x  c⊤xs.t.Ax+s=b,    s∈K\min_{\mathbf{x}} \; \mathbf{c}^\top\mathbf{x} \quad \text{s.t.} \quad A\mathbf{x} + \mathbf{s} = \mathbf{b}, \;\; \mathbf{s} \in \mathcal{K}
QuantityValue
Variables (n)4
Constraints (m)7
nnz(A)7
Density25.0%
Cone product:
K=R+3×Q4\mathcal{K} = \mathbb{R}_+^{3} \times \mathcal{Q}^{4}

Variable Mapping

Solver indicesNameOrigin
x[1]aux_10auxiliary (canonicalization)
x[2:4]xuser variable (3x1)

Block Structure of A

rows 1–33 rows · inequalitiesR+3\mathbb{R}_+^{3}
rows 4–74 rows · SOC(4)Q4\mathcal{Q}^{4}

Solver Data

Data as seen by CLARABEL
Ax+s=b,s∈KA\mathbf{x} + \mathbf{s} = \mathbf{b}, \quad \mathbf{s} \in \mathcal{K}

Matrix Summaries

A
Shape: 7 × 4
nnz: 7 (25.0%)
b
Type: vector, length 7
nnz: 3
Range: [-1.000, 0.000]
c
Type: vector, length 4
nnz: 1
Range: [0.000, 1.000]

Sparsity Pattern of A