Mehrotra's predictor–corrector method in optimization is a specific interior point method for linear programming. It was proposed in 1989 by Sanjay Mehrotra. 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. 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.
Derivation The derivation of this section follows the outline by Nocedal and Wright.
Predictor step - Affine scaling direction A linear program can always be formulated in the standard form
min x q ( x ) = c T x , s.t. A x = b , x ≥ 0 , {\displaystyle {\begin{aligned}&{\underset {x}{\min }}&q(x)&=c^{T}x,\\&{\text{s.t.}}&Ax&=b,\\&\;&x&\geq 0,\end{aligned}}}
where c ∈ R n × 1 , A ∈ R m × n {\displaystyle c\in \mathbb {R} ^{n\times 1},\;A\in \mathbb {R} ^{m\times n}} and b ∈ R m × 1 {\displaystyle b\in \mathbb {R} ^{m\times 1}} define the problem with m {\displaystyle m} constraints and n {\displaystyle n} equations while x ∈ R n × 1 {\displaystyle x\in \mathbb {R} ^{n\times 1}} is a vector of variables. The Karush-Kuhn-Tucker (KKT) conditions for the problem are
A T λ + s = c , (Lagrange gradient condition) A x = b , (Feasibility condition) X S e = 0 , (Complementarity condition) ( x , s ) ≥ 0 , {\displaystyle {\begin{aligned}A^{T}\lambda +s&=c,\;\;\;{\text{(Lagrange gradient condition)}}\\Ax&=b,\;\;\;{\text{(Feasibility condition)}}\\XSe&=0,\;\;\;{\text{(Complementarity condition)}}\\(x,s)&\geq 0,\end{aligned}}}
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