Mehrotra's predictor–corrector method in optimization is a specific interior point method for linear programming. It was proposed in 1989 by Sanjay Mehrotra.[1]
The method is based on the fact that at each iteration of an interior point algorithm it is necessary to compute the Cholesky decomposition (factorization) of a large matrix to find the search direction. The factorization step is the most computationally expensive step in the algorithm. Therefore, it makes sense to use the same decomposition more than once before recomputing it.
At each iteration of the algorithm, Mehrotra's predictor–corrector method uses the same Cholesky decomposition to find two different directions: a predictor and a corrector.
The idea is to first compute an optimizing search direction based on a first order term (predictor). The step size that can be taken in this direction is used to evaluate how much centrality correction is needed. Then, a corrector term is computed: this contains both a centrality term and a second order term.
The complete search direction is the sum of the predictor direction and the corrector direction.
Although there is no theoretical complexity bound on it yet, Mehrotra's predictor–corrector method is widely used in practice.[2] Its corrector step uses the same Cholesky decomposition found during the predictor step in an effective way, and thus it is only marginally more expensive than a standard interior point algorithm. However, the additional overhead per iteration is usually paid off by a reduction in the number of iterations needed to reach an optimal solution. It also appears to converge very fast when close to the optimum.
The derivation of this section follows the outline by Nocedal and Wright.[3]
A linear program can always be formulated in the standard form
\begin{align} &\underset{x}{min}&q(x)&=cTx,\\ &s.t.&Ax&=b,\\ & &x&\geq0, \end{align}
where
c\inRn, A\inRm
b\inRm
m
n
x\inRn
The Karush-Kuhn-Tucker (KKT) conditions for the problem are
\begin{align} ATλ+s&=c, (Lagrangegradientcondition)\\ Ax&=b, (Feasibilitycondition)\\ XSe&=0, (Complementaritycondition)\\ (x,s)&\geq0, \end{align}
where
X=diag(x)
S=diag(s)
e=(1,1,...,1)T\inRn
These conditions can be reformulated as a mapping
F:R2n+m → R2n+m
\begin{align} F(x,λ,s)=\begin{bmatrix}ATλ+s-c\\Ax-b\\XSe\end{bmatrix}&=0\\ (x,s)&\geq0 \end{align}
The predictor-corrector method then works by using Newton's method to obtain the affine scaling direction. This is achieved by solving the following system of linear equations
J(x,λ,s)\begin{bmatrix}\Deltaxaff\\\Deltaλaff\\\Deltasaff\end{bmatrix}=-F(x,λ,s)
where
J
J(x,λ,s)=\begin{bmatrix}\nablaxF&\nablaλF&\nablasF\end{bmatrix},
is the Jacobian of F.
Thus, the system becomes
\begin{bmatrix}0&AT&I\ A&0&0\ S&0&X\end{bmatrix}\begin{bmatrix}\Deltaxaff\\\Deltaλaff\\\Deltasaff\end{bmatrix}=\begin{bmatrix}-rc\\-rb\\-XSe\end{bmatrix}, rc=ATλ+s-c, rb=Ax-b
The average value of the products
xisi, i=1,2,...,n
(xk,sk)
k
| \mu= | 1 |
| n |
| n | |
| \sum | |
| i=1 |
xisi=
| xTs | |
| n |
.
For a value of the centering parameter,
\sigma\in[0,1],
\begin{bmatrix}0&AT&I\ A&0&0\ S&0&X\end{bmatrix} \begin{bmatrix}\Deltaxcen\\\Deltaλcen\\\Deltascen\end{bmatrix} =\begin{bmatrix}-rc\\-rb\\-XSe+\sigma\mue\end{bmatrix}
Considering the system used to compute the affine scaling direction defined in the above, one can note that taking a full step in the affine scaling direction results in the complementarity condition not being satisfied:
\left(xi+\Delta
| aff\right)\left(s | |
| x | |
| i+\Delta |
| aff\right) | |
| s | |
| i |
=xisi+xi\Delta
| aff | |
| s | |
| i |
+si\Delta
| aff | |
| x | |
| i |
+\Delta
| aff\Delta | |
| x | |
| i |
| aff | |
| s | |
| i |
=\Delta
| aff\Delta | |
| x | |
| i |
| aff | |
| s | |
| i |
\ne0.
As such, a system can be defined to compute a step that attempts to correct for this error. This system relies on the previous computation of the affine scaling direction.
\begin{bmatrix}0&AT&I\ A&0&0\ S&0&X\end{bmatrix} \begin{bmatrix}\Deltaxcor\\\Deltaλcor\\\Deltascor\end{bmatrix} =\begin{bmatrix}0\\0\\-\DeltaXaff\DeltaSaffe\end{bmatrix}
The predictor, corrector and centering contributions to the system right hand side can be aggregated into a single system. This system will depend on the previous computation of the affine scaling direction, however, the system matrix will be identical to that of the predictor step such that its factorization can be reused.
The aggregated system is
\begin{bmatrix}0&AT&I\ A&0&0\ S&0&X\end{bmatrix} \begin{bmatrix}\Deltax\\\Deltaλ\\\Deltas\end{bmatrix} =\begin{bmatrix}-rc\\-rb\\-XSe-\DeltaXaff\DeltaSaffe+\sigma\mue\end{bmatrix}
The predictor-corrector algorithm then first computes the affine scaling direction. Secondly, it solves the aggregated system to obtain the search direction of the current iteration.
The affine scaling direction can be used to define a heuristic to adaptively choose the centering parameter as
\sigma=\left(
| \muaff | |
| \mu |
\right)3,
where
\begin{align} \muaff&=
| pri | |
| (x+\alpha | |
| aff\Delta |
xaff)T(s+\alpha
| dual | |
| aff\Delta |
saff)/n,\\ \alpha
| pri | |
| aff |
&=min\left(1,\underset{i:\Delta
| aff<0}{min} | ||
| x | - | |
| i |
| xi | ||||||||
|
| dual | |
| \right),\\ \alpha | |
| aff |
&=min\left(1,\underset{i:\Delta
| aff<0}{min} | ||
| s | - | |
| i |
| si | ||||||||
|
\right), \end{align}
Here,
\muaff
\mu
In practical implementations, a version of line search is performed to obtain the maximal step length that can be taken in the search direction without violating nonnegativity,
(x,s)\geq0
Although the modifications presented by Mehrotra were intended for interior point algorithms for linear programming, the ideas have been extended and successfully applied to quadratic programming as well.