Consider the case of a convex quadratic objective function, where the constraints are affine i.e.,
minx∈Rd12xTQx+cTx
subject to Ax≤b
Where A∈Rm×d,b∈Rm and c∈Rd. The square symmetric matrix Q∈Rd×d is positive definite, and therefore the objective function is convex.This is known as a quadratic program. Observe that it has d variables and m linear constraints.
Consider the quadratic program of two variables
minx∈R212[x1x2]T[2114][x1x2]+[53]T[x1x2]subject to
[10−10010−1][x1x2]≤[1111]
This is illustrated in figure
The objective function is quadratic with a positive semidefinite matrix Q, resulting in elliptical contour lines. The optimal value must lie in the shaded (feasible) region, and is indicated by the star.
The Lagrangian is given by
L(x,λ)=12xTQx+cTx+λT(Ax−b)
=12xTQx+(c+ATλ)Tx−λTb
where again, we have re arranged the terms.Taking the derivative of L(x,λ) with respect to x and setting it to zero gives
Qx+(c+ATλ)=0
Assuming that Q is invertible, we get
x=−Q−1(c+ATλ)Substituting into the primal Lagrangian L(x,λ), we get the dual Lagrangian
D(λ)=−12(c+ATλ)TQ−1(c+ATλ)−λTb
Therefore the dual optimization problem is given by
maxλ∈Rm−12(c+ATλ)TQ−1(c+ATλ)−λTb
subject to λ≥0
Example: Consider the above problem
Provide necessary and sufficient conditions under which a quadratic optimization problem be written as a linear least squares problem.
Solution
Consider minx∈Rn12xTQx+gTx, then the necessary and sufficient condition is Q is PSD and g∈Ran(Q). Indeed, if Q is PSD, then Q has a Cholesky factorization Q=LLT where L∈Rn×k with k=rank(Q). Since g∈Ran(Q)=Ran(L), there is a vector b∈Rk such that −g=Lb.
Then 12xTQx+gTx=12xTLLTx−(Lb)Tx
=||12(LTx)T(LTx)−bT(LTx)+12bTb||−12bTb
=12||LTx−b||T2−12bTb
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