maximize · 1 variable · 1 constraint

Original Problem

maximize⁡1⊤Entr(x)\underset{}{\operatorname{maximize}} \quad \mathbf{1}^\top {\texttt{Entr}({\mathbf{x}})}
subject to\text{subject to}
1⊤x=1\mathbf{1}^\top {\mathbf{x}} = 1
✓Problem is DCP compliant.
1 variable, 1 constraint
t2=φΣ(t1)t_{2} = \varphi^{\Sigma}({t_{1}})
concave⋅?⋅1×1\text{concave} \cdot ? \cdot 1×1
t1=φEntr(x)t_{1} = \varphi^{\texttt{Entr}}({\mathbf{x}})
concave⋅?⋅5×1\text{concave} \cdot ? \cdot 5×1
x\mathbf{x}
affine⋅?⋅5×1\text{affine} \cdot ? \cdot 5×1
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Constraints

[1] Equality 1×1
t3=φΣ(x)t_{3} = \varphi^{\Sigma}({\mathbf{x}})[affine, 1×1]
x\mathbf{x}[affine, 5×1]
11[constant, 1×1]

Smith Form

maximize⁡t2\operatorname{maximize} \quad t_{2}
subject to:
(1)
t1=φEntr(x)t_{1} = \varphi^{\texttt{Entr}}({\mathbf{x}})
(2)
t2=φΣ(t1)t_{2} = \varphi^{\Sigma}({t_{1}})
(3)
t3=φΣ(x)t_{3} = \varphi^{\Sigma}({\mathbf{x}})
constraints:
[1]
Equality1×1\text{Equality} \quad 1\times 1

Relaxed Smith Form

maximize⁡t2\operatorname{maximize} \quad t_{2}
subject to:
(1)
t1≤φEntr(x)t_{1} \leq \varphi^{\texttt{Entr}}({\mathbf{x}})
(2)
t2=1⊤t1t_{2} = \mathbf{1}^\top {t_{1}}
(3)
t3=1⊤xt_{3} = \mathbf{1}^\top {\mathbf{x}}
constraints:
[1]
Equality1×1\text{Equality} \quad 1\times 1

Conic Form

maximize⁡t2\operatorname{maximize} \quad t_{2}
subject to:
(1)
t1≤φEntr(x)[conic form: see canonicalizer]t_{1} \leq \varphi^{\texttt{Entr}}({\mathbf{x}}) \quad \text{[conic form: see canonicalizer]}
(2)
t2=1⊤t1t_{2} = \mathbf{1}^\top {t_{1}}
(3)
t3=1⊤xt_{3} = \mathbf{1}^\top {\mathbf{x}}
constraints:
[1]
Equality1×1\text{Equality} \quad 1\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)10
Constraints (m)16
nnz(A)15
Density9.4%
Cone product:
K={0}1×Kexp⁡5\mathcal{K} = \{0\}^{1} \times \mathcal{K}_{\exp}^{5}

Variable Mapping

Solver indicesNameOrigin
x[1:5]aux_123auxiliary (canonicalization)
x[6:10]xuser variable (5x1)

Block Structure of A

rows 1–11 rows · equalities{0}1\{0\}^{1}
rows 2–1615 rows · ExpCone x5Kexp⁡5\mathcal{K}_{\exp}^{5}

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: 16 × 10
nnz: 15 (9.4%)
b
Type: vector, length 16
nnz: 6
Range: [0.000, 1.000]
c
Type: vector, length 10
nnz: 5
Range: [-1.000, 0.000]

Sparsity Pattern of A