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Linear Algebraic Optimization

Many optimization problems can be solved using the machinery of Linear Algebra. These problems do not have inequality constraints or non-euclidean norms in the objective function.
Projection
The idea behind projection is to find the closest point in a set closest (with respect to particular norm) to a given point.
Definition 28
Given a vector x\mathbf{x}x
 in inner product space X\mathcal{X}X
 and a subspace S⊆XS\subseteq\mathcal{X}S⊆X
, the projection of x\mathbf{x}x
 onto SSS
 is given by
​ΠS(x)=argminy∈S∥y−x∥\Pi_S(\mathbf{x}) = \text{argmin}_{\mathbf{y}\in S}\|\mathbf{y}-\mathbf{x}\|ΠS​(x)=argminy∈S​∥y−x∥
​
where the norm is the one induced by the inner product.
Theorem 9
There exists a unique vector x∗∈S\mathbf{x}^*\in Sx∗∈S
 which solves
​min⁡y∈S∥y−x∥.\min_{\mathbf{y}\in S} \|\mathbf{y}-\mathbf{x}\|.miny∈S​∥y−x∥.
​
It is necessary and sufficient for x∗\mathbf{x}^*x∗
 to be optimal that (x−x∗)⊥S(\mathbf{x}-\mathbf{x}^*)\perp S(x−x∗)⊥S
. The same condition applies when projecting onto an affine set.
Matrix Pseudo-inverses
Definition 29
A pseudoinverse is a matrix A†A^{\dagger}A†
 that satisfies:
​AA†A=AA†AA†=A†(AA†)T=AA†(A†A)T=A†AA A^\dagger A = A \quad A^\dagger A A^\dagger = A^\dagger \quad (AA^\dagger)^T = A A^\dagger \quad (A^\dagger A)^T = A^\dagger AAA†A=AA†AA†=A†(AA†)T=AA†(A†A)T=A†A
​
There are several special cases of pseudoinverses.
1.​A†=Vrdiag(1σ1,⋯ ,1σr)UrTA^\dagger = V_r \text{diag}\left(\frac{1}{\sigma_1},\cdots,\frac{1}{\sigma_r}\right)U_r^TA†=Vr​diag(σ1​1​,⋯,σr​1​)UrT​
 is the Moore-Penrose Pseudo-inverse.
2.When AAA
 and non-singular, A†=A−1A^\dagger = A^{-1}A†=A−1
.
3.When AAA
 is full column rank, A†=(ATA)−1ATA^\dagger = (A^TA)^{-1}A^TA†=(ATA)−1AT
.
4.When AAA
 is full row rank, A†=AT(AAT)−1A^{\dagger} = A^T(AA^T)^{-1}A†=AT(AAT)−1
​
The pseudo-inverses are useful because they can easily compute the projection of a vector onto a related subspace of AAA
.
1.​argminz∈R(A)∥z−y∥2=AA†y\text{argmin}_{z\in\mathcal{R}(A)}\|\mathbf{z}-\mathbf{y}\|_2 = AA^\dagger \mathbf{y}argminz∈R(A)​∥z−y∥2​=AA†y
​
2.​argminz∈R(A)⊥∥z−y∥2=(I−AA†)y\text{argmin}_{z\in\mathcal{R}(A)^\perp}\|\mathbf{z}-\mathbf{y}\|_2 = (I - AA^\dagger)\mathbf{y}argminz∈R(A)⊥​∥z−y∥2​=(I−AA†)y
​
3.​argminz∈N(A)∥z−y∥2=(I−A†A)y\text{argmin}_{z\in\mathcal{N}(A)}\|\mathbf{z}-\mathbf{y}\|_2 = (I - A^\dagger A)\mathbf{y}argminz∈N(A)​∥z−y∥2​=(I−A†A)y
​
4.​argminz∈N(A)⊥∥z−y∥2=A†Ay\text{argmin}_{z\in\mathcal{N}(A)^\perp}\|\mathbf{z}-\mathbf{y}\|_2 = A^\dagger A\mathbf{y}argminz∈N(A)⊥​∥z−y∥2​=A†Ay
​
Explained Variance
The Low Rank Approximation problem is to approximate a matrix AAA
 with a rank kkk
 matrix
​min⁡Ak∥A−Ak∥F2 such that rank(Ak)=k.\min_{A_k} \|A - A_k\|_F^2 \text{ such that rank}(A_k) = k.minAk​​∥A−Ak​∥F2​ such that rank(Ak​)=k.
​
The solution to the low rank approximation problem is simply the first kkk
 terms of the SVD:
​AK⋆=∑i=1kσiuiviT.A_K^\star = \sum_{i=1}^k \sigma_i\mathbf{u}_i\mathbf{v}^T_i.AK⋆​=∑i=1k​σi​ui​viT​.
​
This is because the singular values give us a notion of how much of the Frobenius Norm (Total Variance) each dyad explains.
​η=∥Ak∥F2∥A∥F2=∑ikσi2∑irσi2\eta = \frac{\|A_k\|_F^2}{\|A\|_F^2} = \frac{\sum_i^k \sigma_i^2}{\sum_i^r \sigma_i^2}η=∥A∥F2​∥Ak​∥F2​​=∑ir​σi2​∑ik​σi2​​
​
PCA
Suppose we had a matrix containing mmm
 data points in Rn\mathbb{R}^nRn
 (each data point is a column), and without loss of generality, assume this data is centered around 0 (i.e ∑ixi=0\sum_i \mathbf{x}_i = 0∑i​xi​=0
). The variance of this data along a particular direction z\mathbf{z}z
 is given by zTXXTz\mathbf{z}^TXX^T\mathbf{z}zTXXTz
. Principle Component Analysis is finding the directions z\mathbf{z}z
 such that the variance is maximized.
​max⁡z∈RnzTXXTz such that ∥z∥2=1\max_{z\in\mathbb{R}^n} \mathbf{z}^TXX^T\mathbf{z} \text{ such that } \|\mathbf{z}\|_2 = 1maxz∈Rn​zTXXTz such that ∥z∥2​=1
​
The left singular vector corresponding to the largest singular value of the XXTXX^TXXT
 matrix is the optimizer of this problem, and the variance along this direction is σ12\sigma_1^2σ12​
. If we wanted to find subsequent directions of maximal variance, they are just the left singular vectors corresponding to the largest singular values.
Removing Constraints
Following from the Fundmental Theorem of Linear Algebra, if Ax=yA\mathbf{x}=\mathbf{y}Ax=y
 has a solution, then the set of solutions can be expressed as
​S={xˉ+Nz}S = \{\bar{\mathbf{x}} + N\mathbf{z}\}S={xˉ+Nz}
​
where Axˉ=yA\bar{\mathbf{x}}=\mathbf{y}Axˉ=y
 and NNN
 is a basis for N(A)\mathcal{N}(A)N(A)
. This means if we have a constrained optimization problem
​min⁡xf0(x) : Ax=b,\min_\mathbf{x} f_0(\mathbf{x}) \ : \ A\mathbf{x} = \mathbf{b},minx​f0​(x) : Ax=b,
​
we can write an equivalent unconstrained problem
​min⁡zf0(x0+Nz)\min_\mathbf{z} f_0(\mathbf{x}_0 + N\mathbf{z})minz​f0​(x0​+Nz)
​
where Ax0=bA\mathbf{x}_0 = \mathbf{b}Ax0​=b
​
Fundamentals of Optimization
Convex Optimization
Last modified 2yr ago