What are the most important algorithms for solving linear programming problems in OR?

Combinatorial optimization and linear programming have always been my go-to methods when dealing with complex decision-making. I like to mix those with simulation techniques to formulate a robust decision-space that can be explored visually in various 2D/3D ways to help decision-makers decide on strategic direction, then refine the process to help them arrive at tactically sufficient solutions.

The Simplex Method remains the dominant workhorse for solving linear programming (LP) problems in Operations Research (OR). Here's why: 1. Versatility and Efficiency: The Simplex Method handles a wide range of LP problems, from small-scale to large-scale, with proven efficiency for well-conditioned problems. 2. Ease of Implementation: The algorithm is relatively simple to understand and implement, making it accessible to both seasoned professionals and those new to LP. 3. Integration with AI and Big Data: While emerging technologies like AI and Big Data hold promise for complex optimization problems, they often rely on the Simplex Method as a fundamental building block.

The simplex method is still one the most common used optimization techniques for linear problems. The method provides a systematic and intuitive way to find optimal solutions to linear problems using and interactive process. Understanding the Simplex is fundamental to understanding other types of optimization methods, such as branch and bound (largely used for Mixed Integer Linear Problems - MIP). One of the advantages of using the simplex method is allowing solving in an efficiently manner problems with a considerable number of decision variables and constraints. Linear programming remains one of the main tools for optimizing processes and reducing costs, even for the most recent industries and segments.

The simplex method is a powerful optimization technique used in artificial intelligence to solve linear programming problems. It iteratively improves a feasible solution until it reaches an optimal one by traversing along the edges of a feasible region. Through a series of steps, it adjusts variables to maximize or minimize an objective function while satisfying constraints. Its efficiency and effectiveness make it widely used in AI for resource allocation, production planning, and other decision-making processes.

The most important algorithms for solving linear programming problems in Operations Research include the Simplex Method, Interior Point Methods, Dual Simplex Method, Primal-Dual Interior Point Methods, Barrier Methods, Network Flow Algorithms, Ellipsoid Method, and Benders Decomposition. The choice depends on problem characteristics like size and feasibility.

The Interior Point Method has gained popularity in the field of Operations Research due to its notable efficiency across various problem instances, particularly in handling large-scale optimization problems with a high degree of accuracy and computational effectiveness.

Interior point methods are in solving the complex optimization problems such as multi objective, multi disciplinary equations. This has distinct advantage in solving linear and nonlinear convex optimization problems. In my experience I found using IP methods are great. Indian Mathematician "Narendra Karmarkar" contributed greatly to this, solving linear programming problems in polynomial time.

Is a more recent method developed at Bell labs to solver very large network problems. Ideally suited for large LP problems. Most commercial packages have thus method, some automatically switch between simplex and interior point method depending on the problem complexity.

Interior point methods are well-suited to solve unconstrained problems because of their ability to handle inequality constraints very efficiently by using the logarithmic barrier function

In theory, interior-point methods are considered superior to simplex methods because of their polynomial time complexity, while simplex methods can exhibit exponential runtime in the worst-case scenario. However, despite this theoretical advantage, simplex methods have proven to be significantly more practical and easier to implement. Interior-point methods navigate through the interior of the feasible region to attain the optimal solution and are applicable for solving both linear and nonlinear optimization problems.

Branch and bound remain more effective tool for Integer Linear Programming or Mixed Integer Programming Problems. However, most times, these finding oprimql solutions could be computational expensive. The practice however invlove relaxing the binary constraint during computational experiments. This is because the nature of ILP and MILP which is complex and complicated.

Pretty much the standard method for Mixed Integer LP problems. The computer time could double with each additional binary variable, but that is the absolute worst case. However such is not the case for well formulated real life (not contrived) problems. I have routinely solved problems with dozens of binary variables within a second and within minutes for problems with binaries approaching 100. My advice, do not hesitate to use them but judiciously.

Branch and bound (B&B) method is kind of the extension of method which deals with integer value as the solution. B&B is initially developed to deal with ILP. However, in my experience, B&B method could also be implemented on MILP (Mixed Integer Linear Programming) problem as well since this kind of problem also deals with integer value solution. In my experience as well, B&B may require longer computational time to find the solution even when you are exploiting optimization software. So, if you are dealing with either ILP or MILP problem, you may be ready with this type of drawback.

Cutting plane method maybe very useful for someone who is really into visual thing to understand how optimization problems can be solved. This will help you to find the solution with given feasible area bounded by constraints expressed by lines. However, you would need to be so careful when defining where the feasible area is by carefully looking at the sign of the constraints either <, > , <=, >=, or even = since it's pretty crucial.

cutting plane methods solve nonlinear optimization problems by converting into linear equations. The obtained optimum is tested for being an integer solution. If it is not, there is guaranteed to exist a linear inequality that separates the optimum from the convex hull of the true feasible set.

This method is m I re often used for Pure Integer LP problems, however my experience has been limited in this arena. Some commercial software automatically choose between this and B&B based on problem characteristics.

Some commercial software, my experience has been with Lindo and Lingo, do a wonderful job of exploiting special structures in the problem and thus reducing the compute time significantly. This becomes an important consideration for large scale problems.

Decomposition method translates a constraint satisfaction problem into another constraint satisfaction problem. Decomposition methods translate a problem into a new one that is easier to solve. In Lagrangian method, complex constraints problem is modified to simpler constraints problems which are easy to solve. Lagrange multiplier are used in converting the equation. In ALE method additional penalty term is added.

If your regular model involves very large number (more than few thousands) of constraints then it is advised to use either Decomposition based methods or more complex, in terms of iterations and memory requirements, the interior-point methods, such as Karmarkar's.

Decomposition Algorithms play a crucial role in tackling intricate linear programming challenges by partitioning them into smaller, more tractable subproblems. By strategically decomposing the original problem, these algorithms enable efficient optimization strategies and facilitate the utilization of specialized solution techniques tailored to each subproblem. This approach not only enhances computational efficiency but also fosters a clearer understanding of the problem structure, ultimately leading to improved overall solution quality.

The Simplex method and the Interior Point method are crucial for solving linear programming problems. The Simplex method is efficient for practical applications, while the Interior Point method is better suited for large-scale problems due to its polynomial time complexity (though in practice they have similar runtimes). For integer programming, the Branch and Bound algorithm and the Cutting Plane method are key. Branch and Bound efficiently narrows the search for the optimal solution by exploring subproblems, while the Cutting Plane method progressively refines the solution space with additional constraints.

The elephant in the room is, of course, that hardly anyone has deterministic problems to solve. We keep pretending that the uncertainty did not matter anymore once we have mean predictions from our M department, but this is simply not true. That is why the reader should consider other, primal methods for tackling real-world optimization problems. One such method is hyper-reactive search, which is essentially local search that combines recombination and interpolation while *learning* offline how runtime search features should be leveraged to guide the search online. This is the approach that powers solvers like InsideOpt Seeker.

For LP it is simplex and interior-point. Other algorithms like branch and bound, cutting plane, brand and cut etc. are used to get integer solutions.

There are several considerations for selecting appropriate linear programming techniques, each with specific characteristics, benefits and limitations. For instance, simplex offers efficiency and interpretability for sparse problems, and interior point methods provide scalability and guaranteed time performance for dense problems. Understanding the different method types is essential to select effective and computationally efficient solutions. The primary challenge, as has been discussed, is that often, the underlying data and problem are non-deterministic and must be optimised under uncertainty. Fundamental challenges arise in linear programming for this case.

The graphical method can also be considered in solving linear programming problems, however it is mostly used for simple equations