This project has two (2) sections, the ALM Algorithm for the pressure vessel design optimization and the PSO for the welded beam design.

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Section 1: Augmented Lagrangian Multiplier (ALM) Algorithm for Pressure Vessel design.

• The ALM algorithm is an indirect method for constrained optimization problems, the constrained optimization problem is transformed into an unconstrained optimization problem by forming a composite function using the cost, constraint functions and penalty parameters. 

• The transformed or composite unconstrained function was then solved using a Quasi-Newton Method, the Davidon-Fletcher-Powell (DFP) method. This
  method was chosen because it is one of the most powerful methods for minimization of a function due to its fast rate of convergence compared to the ordinary steepest-descent method, its robustness and computational efficiency.

• The Golden search method was used to used to solve the step-size determination subproblem in the DFP subroutine.

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Section 2: Particle Swarm Optimization Algorithm for the Welded beam design.

• Particle swarm optimization (PSO) is inspired by social and cooperative behavior displayed by various species to fulfill their needs in the search space. The algorithm is guided by personal experience (Pbest), overall experience (Gbest) and the present movement of the particles to decide their next positions in the search space.

• For constrained optimization problems, just like indirect search methods, an equivalent unconstrained function is constructed using a penalty function for the constraints and the PSO algorithm is applied.

• In literature, the stationary penalty parameters have proved to work better than the non-stationary penalty, so it was applied in this project. Also, the most commonly used hyperparameters for the PSO (not mathematically proven but a rule-of-thumb),
  these hyperparameters have worked well with the PSO algorithm and they were adopted in this project.

• For improvement of the particle swarm optimization algorithm, an inertia weight and constriction factor was added to reduce the velocity value because the particles' velocities tend to build up so fast which might lead to skipping the optimum point.

• Finally, boundary conditions were introduced to the problem, this is to confine the search space and prevent the particle from going to a position that will result in invalid solution.

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