Optimization algorithms can handle non-linear objective functions through various techniques. Non-linear objective functions are mathematical functions that are not linear in nature, often involving terms with higher powers or non-linear mathematical operations such as exponentials, logarithms, or trigonometric functions.

  One commonly used approach is to employ iterative optimization algorithms such as gradient descent, Newton's method, or the Levenberg-Marquardt algorithm. These algorithms aim to find the minimum (or maximum) of the non-linear objective function by iteratively updating the solution towards the optimal point. They achieve this by computing the gradient or the Hessian matrix of the objective function, which indicates the direction and rate of change at a particular point.

  Gradient descent is an optimization algorithm that iteratively adjusts the solution in the direction of steepest descent of the objective function. It computes the gradient of the objective function at each iteration and updates the solution by taking a step in the opposite direction of the gradient. This process continues until a stopping criterion is met, such as reaching a predefined tolerance or a maximum number of iterations.

  Newton's method, on the other hand, uses the second-order derivative of the objective function (the Hessian matrix) in addition to the gradient. It approximates the objective function locally by a quadratic function and then finds the minimum of this approximation. Newton's method often converges faster than gradient descent but may require more computational resources due to the calculation of the Hessian matrix.

  The Levenberg-Marquardt algorithm is commonly used for non-linear least squares problems. It is an iterative optimization algorithm that performs a combination of gradient descent and Gauss-Newton iterations. It effectively balances between a steepest descent direction and a curvature-based direction to find the minimum of the objective function.

  There are also metaheuristic algorithms that can handle non-linear objective functions, such as genetic algorithms, simulated annealing, particle swarm optimization, and ant colony optimization. These algorithms are based on probabilistic or nature-inspired techniques and are suitable for solving complex optimization problems with non-linear objectives and possibly non-linear constraints. They explore the solution space and iteratively improve the solutions until reaching an optimal or near-optimal solution.

  Overall, the choice of optimization algorithm depends on the specific characteristics of the non-linear objective function, the available computational resources, and the desired tradeoff between solution quality and computational time.