minimize · 1 variable · 0 constraints

Original Problem

minimize⁡∥C4×3x+−C4×1∥221\underset{}{\operatorname{minimize}} \quad \frac{\lVert {{{C_{4\times 3}} {\mathbf{x}}} + {-{C_{4\times 1}}}} \rVert_2^2}{{1}}
✓Problem is DCP compliant.
1 variable, 0 constraints
t4=φqol(t3,1)t_{4} = \varphi^{\text{qol}}({t_{3}}, {1})
convex⋅R+⋅1×1\text{convex} \cdot \mathbb{R}_+ \cdot 1×1
t3=t1+t2t_{3} = {t_{1}} + {t_{2}}
affine⋅?⋅4×1\text{affine} \cdot ? \cdot 4×1
11
constant⋅R+⋅1×1\text{constant} \cdot \mathbb{R}_+ \cdot 1×1
t1=C4×3xt_{1} = {C_{4\times 3}} {\mathbf{x}}
affine⋅?⋅4×1\text{affine} \cdot ? \cdot 4×1
t2=−C4×1t_{2} = -{C_{4\times 1}}
constant⋅?⋅4×1\text{constant} \cdot ? \cdot 4×1
C4×3C_{4\times 3}
constant⋅?⋅4×3\text{constant} \cdot ? \cdot 4×3
x\mathbf{x}
affine⋅?⋅3×1\text{affine} \cdot ? \cdot 3×1
C4×1C_{4\times 1}
constant⋅?⋅4×1\text{constant} \cdot ? \cdot 4×1
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Smith Form

minimize⁡t4\operatorname{minimize} \quad t_{4}
subject to:
(1)
t1=C4×3xt_{1} = {C_{4\times 3}} {\mathbf{x}}
(2)
t2=−C4×1t_{2} = -{C_{4\times 1}}
(3)
t3=t1+t2t_{3} = {t_{1}} + {t_{2}}
(4)
t4=φqol(t3,1)t_{4} = \varphi^{\text{qol}}({t_{3}}, {1})

Relaxed Smith Form

minimize⁡t4\operatorname{minimize} \quad t_{4}
subject to:
(1)
t1=C4×3xt_{1} = {C_{4\times 3}} {\mathbf{x}}
(2)
t2=−C4×1t_{2} = -{C_{4\times 1}}
(3)
t3=t1+t2t_{3} = {t_{1}} + {t_{2}}
(4)
t4≥φqol(t3,1)t_{4} \geq \varphi^{\text{qol}}({t_{3}}, {1})

Conic Form

minimize⁡t4\operatorname{minimize} \quad t_{4}
subject to:
(1)
t1=C4×3xt_{1} = {C_{4\times 3}} {\mathbf{x}}
(2)
t2=−C4×1t_{2} = -{C_{4\times 1}}
(3)
t3=t1+t2t_{3} = {t_{1}} + {t_{2}}
(4)
(1+t42,  1−t42,  t3)∈Qn+2\left(\frac{{1}+t_{4}}{2},\; \frac{{1}-t_{4}}{2},\; {t_{3}}\right) \in \mathcal{Q}^{n+2}
(5)
1∈Q1{1} \in \mathcal{Q}^1

Standard Cone Form

min⁡x  12x⊤Px+q⊤xs.t.Aeqx=beq,    Fx≤g\min_{\mathbf{x}} \; \tfrac{1}{2}\mathbf{x}^\top P\mathbf{x} + \mathbf{q}^\top\mathbf{x} \quad \text{s.t.} \quad A_{\text{eq}}\mathbf{x} = \mathbf{b}_{\text{eq}}, \;\; F\mathbf{x} \leq \mathbf{g}
QuantityValue
Variables (n)7
Constraints (m)4
nnz(A)16
Density57.1%
nnz(P)4

Variable Mapping

Solver indicesNameOrigin
x[1:4]aux_35auxiliary (canonicalization)
x[5:7]xuser variable (3x1)

Block Structure of A

rows 1–44 rows · equalities{0}4\{0\}^{4}

Solver Data

Data as seen by OSQP
12x⊤Px+q⊤xs.t.Aeqx=beq,    Fx≤g\tfrac{1}{2}\mathbf{x}^\top P\mathbf{x} + \mathbf{q}^\top\mathbf{x} \quad \text{s.t.} \quad A_{\text{eq}}\mathbf{x} = \mathbf{b}_{\text{eq}}, \;\; F\mathbf{x} \leq \mathbf{g}

Matrix Summaries

P
Shape: 7 × 7
nnz: 4 (8.2%)
q
Type: vector, length 7
nnz: 0
Range: [0.000, 0.000]
A_eq
Shape: 4 × 7
nnz: 16 (57.1%)
b_eq
Type: vector, length 4
nnz: 4
Range: [-1.389, 0.6360]
F_ineq
Shape: 0 × 7
nnz: 0 (0.0%)
g_ineq
Type: vector, length 0
nnz: 0
Range: [0.000, 0.000]

Sparsity Pattern of A