minimize · 1 variable · 1 constraint

Original Problem

minimize⁡Index(x)⊙Index(x)\underset{}{\operatorname{minimize}} \quad {\texttt{Index}({\mathbf{x}})} \odot {\texttt{Index}({\mathbf{x}})}
subject to\text{subject to}
0≤x0 \leq \mathbf{x}
⚠️Problem is NOT DCP compliant. See DCP Analysis tab for details.
1 variable, 1 constraint

DCP Analysis

⚠️Problem is NOT DCP compliant.
✗ Objective (minimize): requires convex expression
Minimize requires a convex objective, but expression is unknown.
✗ Multiply: unknown [1×1]
arg 1: affine, non-monotone (cvx-rule ✓, ccv-rule ✓)
arg 2: affine, non-monotone (cvx-rule ✓, ccv-rule ✓)
⇒ Multiply is neither convex nor concave (atom-level property).
t3=t1⊙t2t_{3} = {t_{1}} \odot {t_{2}}
unknown⋅?⋅1×1\text{unknown} \cdot ? \cdot 1×1
t1=φIndex(x)t_{1} = \varphi^{\texttt{Index}}({\mathbf{x}})
affine⋅?⋅1×1\text{affine} \cdot ? \cdot 1×1
t2=φIndex(x)t_{2} = \varphi^{\texttt{Index}}({\mathbf{x}})
affine⋅?⋅1×1\text{affine} \cdot ? \cdot 1×1
x\mathbf{x}
affine⋅?⋅2×1\text{affine} \cdot ? \cdot 2×1
x\mathbf{x}
affine⋅?⋅2×1\text{affine} \cdot ? \cdot 2×1
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Constraints

[1] Inequality 2×1
00[constant, 1×1]
x\mathbf{x}[affine, 2×1]

Smith Form

minimize⁡t3\operatorname{minimize} \quad t_{3}
subject to:
(1)
t1=φIndex(x)t_{1} = \varphi^{\texttt{Index}}({\mathbf{x}})
(2)
t2=φIndex(x)t_{2} = \varphi^{\texttt{Index}}({\mathbf{x}})
(3)
t3=t1⊙t2t_{3} = {t_{1}} \odot {t_{2}}
constraints:
[1]
Inequality2×1\text{Inequality} \quad 2\times 1

Relaxed Smith Form

minimize⁡t3\operatorname{minimize} \quad t_{3}
subject to:
(1)
t1=φIndex(x)t_{1} = \varphi^{\texttt{Index}}({\mathbf{x}})
(2)
t2=φIndex(x)t_{2} = \varphi^{\texttt{Index}}({\mathbf{x}})
(3)
t3=t1⊙t2t_{3} = {t_{1}} \odot {t_{2}}
constraints:
[1]
Inequality2×1\text{Inequality} \quad 2\times 1

Conic Form

minimize⁡t3\operatorname{minimize} \quad t_{3}
subject to:
(1)
t1=φIndex(x)[conic form: see canonicalizer]t_{1} = \varphi^{\texttt{Index}}({\mathbf{x}}) \quad \text{[conic form: see canonicalizer]}
(2)
t2=φIndex(x)[conic form: see canonicalizer]t_{2} = \varphi^{\texttt{Index}}({\mathbf{x}}) \quad \text{[conic form: see canonicalizer]}
(3)
t3=t1⊙t2t_{3} = {t_{1}} \odot {t_{2}}
constraints:
[1]
Inequality2×1\text{Inequality} \quad 2\times 1