## 5 Multiplicative Iterative Algorithms  
The OT algorithms presented in the last section have the following important achievements: (1) they manage to transform a high dimensional optimization problem into a series of 1-D optimizations; (2) due to 1-D optimizations, the non-negativity constraints can be easily enforced by simply resetting negative estimates to zero in each iteration; (3) the surrogate given by the separable parabola approach is general enough to be applicable to different tomographic reconstructions. A limitation of OT is that it requires all \(l_{i}(\cdot)\) (log-density) and \(-J(\cdot)\) (negative penalty) to be concave functions. In this section we discuss a competitive alternative to the OT method called the multiplicative iterative (MI) algorithm; its application to tomographic imaging can be found in [26] and to box-constrained image processing in [41]. The main motivation of the MI algorithm is that it can be easily derived under different imaging modalities and different measurement noise models. Moreover, for some difficult penalties, such as TV, or even non-convex penalties [42], MI can be easily implemented to solve the corresponding optimization problems. A general MI updating formula can be developed suitable for all tomographic reconstruction problems regardless of the mean function model, measurement probability distribution and penalty function. The simulation study reported in [26] reveals that MI has competitive convergence speed when compared with OT and other reconstruction algorithms. The MI algorithm does not require concavity of the functions \(l_{i}\) and \(-J\) and therefore is more general than the OT algorithm. It requires existence of the first derivatives of \(l_{i}(\cdot)\) and \(J(\cdot)\). It is possible that the objective function \(\Psi(\mathbf{x})\) in Equation (2) has multiple local maxima. In this case, MI finds one of the local non-negative maxima, depending on the starting value of the algorithm. Here are some notations needed to explain the MI algorithm. For a function \(b(z)\), let \({b(z)}^{+}\) be the positive component of \(b(z)\) and \({b(z)}^{-}\) the negative component so that \({b(z)}={b(z)}^{+}+{b(z)}^{-}.\) For a number \(b\), Let \([b]^{+}=\max(0,b)\) and \([b]^{-}=\min(0,b)\) so that \(b=[b]^{+}+[b]^{-}\). Thus, for the numerical value of function \(b(\cdot)\) at point \(z^{*}\), we can also write \(b(z^{*})=[b(z^{*})]^{+}+[b(z^{*})]^{-}.\)  
We develop the MI algorithm from the Karush-Kuhn-Tucker (KKT) necessary conditions for the non-negatively constrained optimization of \(\Psi(x)\). They are:  
\[\
abla_{j}\Psi(\mathbf{x}) =0\text{ if }x_{j}>0\text{ and } \tag{49}\] \[\
abla_{j}\Psi(\mathbf{x}) \leq 0\text{ if }x_{j}=0 \tag{50}\]  
for \(j=1,\ldots,p\). Therefore, we aim to solve for \(\mathbf{x}\) from  
\[x_{j}\left(\sum_{i=1}^{n}\
abla l_{i}(\mu_{i})\
abla_{j}\mu_{i}(\mathbf{x})-h \
abla_{j}J(\mathbf{x})\right)=0 \tag{51}\]  
Note that the expression inside the brackets of Equation (51) represents \(\
abla_{j}\Psi(x)\), and \(x_{j}\) is included in Equation (51) to reflect the conditions in Equations (49) and (50). The key step in developing the MI algorithm is to rearrange Equation (51) such that its positive and negative terms appear on different sides of the Equation (51).