minimize · 1 variable · 0 constraints

Original Problem

minimize⁡max⁡{∣x∣1,…,∣x∣n}+1⊤max⁡(x,0)\underset{}{\operatorname{minimize}} \quad {\max\{{|{\mathbf{x}}|}_1, \ldots, {|{\mathbf{x}}|}_n\}} + {\mathbf{1}^\top {\max({\mathbf{x}}, {0})}}
✓Problem is DCP compliant.
1 variable, 0 constraints
t5=t2+t4t_{5} = {t_{2}} + {t_{4}}
convex⋅R+⋅1×1\text{convex} \cdot \mathbb{R}_+ \cdot 1×1
t2=φmax⁡(t1)t_{2} = \varphi^{\max}({t_{1}})
convex⋅R+⋅1×1\text{convex} \cdot \mathbb{R}_+ \cdot 1×1
t4=φΣ(t3)t_{4} = \varphi^{\Sigma}({t_{3}})
convex⋅R+⋅1×1\text{convex} \cdot \mathbb{R}_+ \cdot 1×1
t1=φ∣⋅∣(x)t_{1} = \varphi^{|\cdot|}({\mathbf{x}})
convex⋅R+⋅3×1\text{convex} \cdot \mathbb{R}_+ \cdot 3×1
t3=φmax⁡(x,0)t_{3} = \varphi^{\max}({\mathbf{x}}, {0})
convex⋅R+⋅3×1\text{convex} \cdot \mathbb{R}_+ \cdot 3×1
x\mathbf{x}
affine⋅?⋅3×1\text{affine} \cdot ? \cdot 3×1
x\mathbf{x}
affine⋅?⋅3×1\text{affine} \cdot ? \cdot 3×1
00
constant⋅zero⋅1×1\text{constant} \cdot zero \cdot 1×1
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Smith Form

minimize⁡t5\operatorname{minimize} \quad t_{5}
subject to:
(1)
t1=φ∣⋅∣(x)t_{1} = \varphi^{|\cdot|}({\mathbf{x}})
(2)
t2=φmax⁡(t1)t_{2} = \varphi^{\max}({t_{1}})
(3)
t3=φmax⁡(x,0)t_{3} = \varphi^{\max}({\mathbf{x}}, {0})
(4)
t4=φΣ(t3)t_{4} = \varphi^{\Sigma}({t_{3}})
(5)
t5=t2+t4t_{5} = {t_{2}} + {t_{4}}

Relaxed Smith Form

minimize⁡t5\operatorname{minimize} \quad t_{5}
subject to:
(1)
t1≥φ∣⋅∣(x)t_{1} \geq \varphi^{|\cdot|}({\mathbf{x}})
(2)
t2≥φmax⁡(t1)t_{2} \geq \varphi^{\max}({t_{1}})
(3)
t3≥φmax⁡(x,0)t_{3} \geq \varphi^{\max}({\mathbf{x}}, {0})
(4)
t4=1⊤t3t_{4} = \mathbf{1}^\top {t_{3}}
(5)
t5=t2+t4t_{5} = {t_{2}} + {t_{4}}

Conic Form

minimize⁡t5\operatorname{minimize} \quad t_{5}
subject to:
(1)
(t1,x)∈Q2(t_{1}, {\mathbf{x}}) \in \mathcal{Q}^2
(2)
t2−t1i∈Q1,  i=1,…,nt_{2} - {t_{1}}_i \in \mathcal{Q}^1,\; i = 1, \ldots, n
(3)
t3≥x,  t3≥0t_{3} \geq {\mathbf{x}},\; t_{3} \geq {0}
(4)
t4=1⊤t3t_{4} = \mathbf{1}^\top {t_{3}}
(5)
t5=t2+t4t_{5} = {t_{2}} + {t_{4}}

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)10
Constraints (m)18
nnz(A)30
Density16.7%
Cone product:
K=R+18\mathcal{K} = \mathbb{R}_+^{18}

Variable Mapping

Solver indicesNameOrigin
x[1]aux_282auxiliary (canonicalization)
x[2:4]aux_288auxiliary (canonicalization)
x[5:7]xuser variable (3x1)
x[8:10]aux_272auxiliary (canonicalization)

Block Structure of A

rows 1–1818 rows · inequalitiesR+18\mathbb{R}_+^{18}

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: 18 × 10
nnz: 30 (16.7%)
b
Type: vector, length 18
nnz: 0
Range: [0.000, 0.000]
c
Type: vector, length 10
nnz: 4
Range: [0.000, 1.000]

Sparsity Pattern of A