Multivariable Optimization with Non-Negativity Constraints

**Problem Statement:** I aim to address a multivariable optimization problem where only solutions with non-negative components are relevant, as negative allocations of resources are not feasible. Approach: One method to converge to the desired solution is by providing an iterative algorithm with a starting point close to the desired solution. If the approximate location of the solution is unknown, a related problem can be solved to generate an initial approximation. For non-negative solutions, variables can be replaced with their square values to ensure positivity. Solution Methodology: The solution employs a two-stage optimization process: 1. **Auxiliary Function Approximation**: An auxiliary function is used to transform the problem into one that guarantees non-negative solutions. 2. **Two-Stage Optimization**: - **Stage 1**: Optimize the auxiliary function to approximate the solution. - **Stage 2**: Refine the approximation by solving the original problem. Example Scenarios: In a two-variable scenario, the problem may have solutions with both positive and negative components. However, only the non-negative solution is of interest. By applying this two-stage process, we can ensure convergence to the desired solution. For higher-dimensional problems, the same process can be extended, ensuring the results align with the constraints of non-negativity.