Aerospace collectively represents one of the most sophisticated technological endeavors and largest markets in the
world. Coming with substantial costs, nearly every aspect of the industry, from aircraft design to material selection
to operation, has been optimized in at least one way. A critical design consideration in any aircraft is...
Optimization and Game Theory have certain conceptual overlaps. It is even said that John von Neumann
conjectured the Duality Theorem using information from his game theory. This article discusses two optimization
applications to the game theory: a methodology for solving the Nash Equilibrium and a decentralized model in
This article concerns the exponential transformation method for globally solving posynomial (or general
geometric/signomial) optimization problems with nonconvex objective functions or constraints. A discussion of the
method's development and use will be presented.
Logarithmic transformation is a method used to change geometric programs into their convex forms. A
geometric program, or GP, is a type of global optimization problem that concerns minimizing a subject to
constraint functions so as to allow one to solve unique non-linear programming problems. All geometric programs
McCormick Envelopes are a type of convex relaxation used in
bilinear Non Linear Programming problems. Many times these
envelopes are used to solve a Mixed Integer Non Linear
Programming problem by relaxing the MINLP problem so that
it becomes a convex NLP. Solving this convex NLP will
provide a lower...
Spatial branch-and-bound is a divide-and-conquer technique used to find the deterministic solution of global optimization
problems.1 It is a type of branch-and-bound method, which solves for the set of parameters that globally optimize the
objective function, whether that be finding the minimum or maximum value of or , respectively, where...
Computational complexity refers to the amount of resources
required to solve a type of problem by systematic application of an
algorithm. Resources that can be considered include the amount of
communications, gates in a circuit, or the number of processors.
Because the size of the particular input to a problem...
The objective of game theory is to analyze the relationship
between decision-making situations in order to achieve a
desirable outcome. The theory can be applied to a wide range
of applications, including, but not limited to, economics,
politics and even the biological sciences. In essence, game
theory serves as means...
Network Flow Optimization problems form the most special class of linear programming problems.
Transportation, electric, and communication networks are clearly common applications of Network Optimization.
These types of problems can be viewed as minimizing transportation problems. This Network problem will include
cost of moving materials through a network involving varying...
Interior point methods are a type of algorithm that are used in
solving both linear and nonlinear convex optimization
problems that contain inequalities as constraints. The LP
Interior-Point method relies on having a linear programming
model with the objective function and all constraints being
continuous and twice continuously differentiable. In...
Optimization with absolute values is a special case of linear programming in which a problem made nonlinear due
to the presence of absolute values is solved using linear programming methods.
Absolute value functions themselves are very difficult to perform standard optimization procedures on. They are
not continuously differentiable functions, are...
Facility location problems deal with selecting the placement of a facility (often from a list of integer possibilities)
to best meet the demanded constraints. The problem often consists of selecting a factory location that minimizes
total weighted distances from suppliers and customers, where weights are representative of the difficulty of...
The traveling salesman problem (TSP) is a widely studied combinatorial optimization problem, which, given a set of cities and a cost to travel from one city to another, seeks to identify the tour that will allow a salesman to visit each city only once, starting and ending in the same...
Mixed-integer cuts or Cutting-plane methods is an iterative approach used to simplify the solution of a mixed
integer linear programming (MILP) problem. Cutting-plane methods work by first relaxing the MILP to a
complementary linear programming problem and cutting the feasible region to narrow down the solution search
space to only...
A disjunctive inequality is a type of constraint that exists in mixed integer linear programming (MILP) and mixed
integer nonlinear programming (MINLP) problems. It involves constraining a solution space with multiple
inequalities or sets of inequalities related by an OR statement. This "OR" statement must then be reformulated
Lagrangian duality theory refers to a way to find a bound or solve an optimization problem (the primal problem) by
looking at a different optimization problem (the dual problem). More specifically, the solution to the dual problem
can provide a bound to the primal problem or the same optimal solution...
Column generation algorithms are used for MILP problems. The formulation was initially proposed by Ford and
Fulkerson in 1958 . The main advantage of column generation is that not all possibilities need to be enumerated.
Instead, the problem is first formulated as a restricted master problem (RMP). This RMP has...
A heuristic algorithm is one that is designed to solve a problem in a faster and more efficient fashion than
traditional methods by sacrificing optimality, accuracy, precision, or completeness for speed. Heuristic algorithms
often times used to solve NP-complete problems, a class of decision problems. In these problems, there is...
Branch and cut method is a very successful algorithm for solving a variety of integer programming problems, and
it also can provide a guarantee of optimality. Many problems involve variables which are not continuous but
instead have integer values, and they can be solved by branch-and cut method. This method...