minimize · 1 variable · 1 constraint

Original Problem

minimize⁡LogSumExp(x)\underset{}{\operatorname{minimize}} \quad \texttt{LogSumExp}({\mathbf{x}})
subject to\text{subject to}
−1≤x-1 \leq \mathbf{x}
✓Problem is DCP compliant.
1 variable, 1 constraint
t1=φLogSumExp(x)t_{1} = \varphi^{\texttt{LogSumExp}}({\mathbf{x}})
convex⋅?⋅1×1\text{convex} \cdot ? \cdot 1×1
x\mathbf{x}
affine⋅?⋅4×1\text{affine} \cdot ? \cdot 4×1
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Constraints

[1] Inequality 4×1
−1-1[constant, 1×1]
x\mathbf{x}[affine, 4×1]

Smith Form

minimize⁡t1\operatorname{minimize} \quad t_{1}
subject to:
(1)
t1=φLogSumExp(x)t_{1} = \varphi^{\texttt{LogSumExp}}({\mathbf{x}})
constraints:
[1]
Inequality4×1\text{Inequality} \quad 4\times 1

Relaxed Smith Form

minimize⁡t1\operatorname{minimize} \quad t_{1}
subject to:
(1)
t1≥φLogSumExp(x)t_{1} \geq \varphi^{\texttt{LogSumExp}}({\mathbf{x}})
constraints:
[1]
Inequality4×1\text{Inequality} \quad 4\times 1

Conic Form

minimize⁡t1\operatorname{minimize} \quad t_{1}
subject to:
(1)
t1≥φLogSumExp(x)[conic form: see canonicalizer]t_{1} \geq \varphi^{\texttt{LogSumExp}}({\mathbf{x}}) \quad \text{[conic form: see canonicalizer]}
constraints:
[1]
Inequality4×1\text{Inequality} \quad 4\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)9
Constraints (m)17
nnz(A)20
Density13.1%
Cone product:
K=R+5×Kexp⁡4\mathcal{K} = \mathbb{R}_+^{5} \times \mathcal{K}_{\exp}^{4}

Variable Mapping

Solver indicesNameOrigin
x[1]aux_196auxiliary (canonicalization)
x[2:5]xuser variable (4x1)
x[6:9]aux_201auxiliary (canonicalization)

Block Structure of A

rows 1–55 rows · inequalitiesR+5\mathbb{R}_+^{5}
rows 6–1712 rows · ExpCone x4Kexp⁡4\mathcal{K}_{\exp}^{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: 17 × 9
nnz: 20 (13.1%)
b
Type: vector, length 17
nnz: 9
Range: [0.000, 1.000]
c
Type: vector, length 9
nnz: 1
Range: [0.000, 1.000]

Sparsity Pattern of A