Sub-iterated Approximation Method：A Solution for Solving Nonlinear Optimization Problems
Since the dawn of civilization, the human race has striven to find effective solutions to optimize resources and increase efficiency. The development of analytical methods, such as calculus and linear algebra, has indeed proven advantageous in solving some optimization problems. Yet, many of the real-world optimization problems confronting us today are nonlinear in nature, and thus, require alternative approaches to numerical optimization. As a result, Sub-iterated Approximation (SIM) method has emerged as a popular solution to the issue of nonlinear optimization.
The objective of the Sub-iterated Approximation Method (SIM) is to reduce a nonlinear optimization problem to two linear programming sub-problems. In this approach, the nonlinear objective function is regenerated into a sorted sequence of quadratic approximations using finite differences. The fact that the new problem transformed into a linear programming problem allows us to utilize existing linear programming algorithms to obtain the optimal solution.
The SIM method has three primary stages to optimize the nonlinear problem. First, a homotopy technique is employed at the initial stage of calculating the root of the system and is referred to as a starting point algorithm. Here, the equation is either iteratively solved in a backward or forward direction. Next, an interior-point algorithm is then utilized to efficiently solve the nonlinear problem by breaking up the objective function into a series of smaller sub-problems. Finally, the resulting linear programming problem is then replaced with a sophisticated nonlinear programming algorithm.
The major advantage of the Sub-iterated Approximation Method is its capability to address large, complex nonlinear optimization problems with greatly reduced computational complexity. Here, the subproblems, that are created as a result of simplifying the nonlinear problem, are much easier to solve. As a result, the model is optimized much faster than traditional approaches to nonlinear optimization, and the large number of variables can be addressed using the efficient linear programming approach.
Another benefit of the Sub-iterated Approximation Method is the ability to customize the solution. This is because the precision of the solution can be adjusted to meet specific user requirements. The more subproblems created, the better the accuracy of the results. Additionally, since the starting point algorithm generates the base equation that formulates the subproblem, the user has the option to customize the algorithm to accommodate further constraints.
The Sub-iterated Approximation Method also offers the advantage of being relatively straightforward to implement. Here, the parameterization of the system of equations reduces the complexity of the nonlinear optimization problem. In addition, Since the algorithm is based upon a nonlinear program, users do not need to worry about developing their own linear programming algorithms.
Although the Sub-iterated Approximation Method provides the means for solving nonlinear optimization challenges quickly, there are some drawbacks. One such disadvantage is that the algorithm can arrive at a solution that does not meet the required level of precision or accuracy. Additionally, the parameterization of the problem can be complex and difficult to comprehend, making it challenging to obtain an accurate solution.
In conclusion, the Sub-iterated Approximation Method is an efficient technique used to solve nonlinear optimization problems. This approach utilizes a series of subproblems, efficiently defined via homotopy and interior-point techniques, that are then translated into a linear programming problem. This allows the user to obtain a solution quickly and efficiently, with the option to customize the resulting solution. Yet, some drawbacks exist in the form of limited precision and an overly complicated parameterization. Nevertheless, SIM is an effective and reasonable solution for addressing nonlinear optimization challenges.
