Robust optimization is a distinct approach to optimizations problems that allows for the incorporation of
uncertainty. The usefulness of robust optimization lies in the ability to solve for every realization of the uncertain
parameters. As a result, the problem can be solved for the worst-case scenarios of the entire set...
Traditionally, robust optimization has solved problems based on static decisions which are predetermined by the
decision makers. Once the decisions were made, the problem was solved and whenever a new uncertainty was
realized, the uncertainty was incorporated to the original problem and the entire problem was solved again to
Robust optimization is a subset of optimization theory that deals with a certain measure of robustness vs uncertainty. This balance of robustness
and uncertainty is represented as variability in the parameters of the problem at hand and or its solution . In robust optimization, the modeler
aims to find decisions...
Fuzzy programming is one of many optimization models that deal with optimization under uncertainty. This model can be applied when situations are not clearly
defined and thus have uncertainty, or an exact value is not critical to the problem. For example, categorizing people into young, middle aged and old is...
The chance-constrained method is one of the major approaches to solving optimization problems under various
uncertainties. It is a formulation of an optimization problem that ensures that the probability of meeting a certain
constraint is above a certain level. In other words, it restricts the feasible region so that the...
Non-differentiable optimization is a category of optimization that deals with objective that for a variety of reasons
is non differentiable and thus non-convex. The functions in this class of optimization are generally non-smooth.
These functions although continuous often contain sharp points or corners that do not allow for the solution...
Geometric programming was introduced in 1967 by Duffin, Peterson and Zener. It is very useful in the applications
of a variety of optimization problems, and falls under the general class of signomial problems. It can be used to
solve large scale, practical problems by quantifying them into a mathematical optimization...
In this work, we will focus on the “at the same time” or direct transcription approach which allow a simultaneous
method for the dynamic optimization problem. In particular, we formulate the dynamic optimization model with
orthogonal collocation methods. These methods can also be regarded as a special class of implicit...
Subgradient Optimization (or Subgradient Method) is an iterative algorithm
for minimizing convex functions, used predominantly in Nondifferentiable
optimization for functions that are convex but nondifferentiable. It is often slower
than Newton's Method when applied to convex differentiable functions, but can
be used on convex nondifferentiable functions where Newton's Method will...
Sequential quadratic programming (SQP) is a class of algorithms for solving non-linear optimization problems
(NLP) in the real world. It is powerful enough for real problems because it can handle any degree of non-linearity
including non-linearity in the constraints. The main disadvantage is that the method incorporates several
Quadratic programming (QP) is the
problem of optimizing a quadratic
objective function and is one of the
simplests form of non-linear
programming. The objective function
can contain bilinear or up to second
order polynomial terms, and the
constraints are linear and can be both
equalities and inequalities. QP is
Quasi-Newton Methods (QNMs) are generally a class of optimization methods that are used in Non-Linear
Programming when full Newton’s Methods are either too time consuming or difficult to use. More specifically,
these methods are used to find the global minimum of a function f(x) that is twice-differentiable. There are distinct...
The conjugate gradient method is a mathematical technique that can be useful
for the optimization of both linear and non-linear systems. This technique is
generally used as an iterative algorithm, however, it can be used as a direct
method, and it will produce a numerical solution. Generally this method is...
The interior point (IP) method for nonlinear programming was pioneered by Anthony V. Fiacco and Garth P. McCormick in the
early 1960s. The basis of IP method restricts the constraints into the objective function (duality
( http://en.wikipedia.org/wiki/Duality_%28optimization%29) ) by creating a barrier function. This limits potential solutions to
iterate in only...
Trust-region method (TRM) is one of the most important numerical optimization methods in
solving nonlinear programming (NLP) problems. It works in a way that first define a region
around the current best solution, in which a certain model (usually a quadratic model) can to
some extent approximate the original objective...
An algorithm is a line search method if it seeks the minimum of a defined nonlinear function by selecting a
reasonable direction vector that, when computed iteratively with a reasonable step size, will provide a function
value closer to the absolute minimum of the function. Varying these will change the...
Extended Cutting Plane is an optimization method suggested by Westerlund and Petersson in 1996 to solve
Mixed-Integer NonLinear Programming (MINLP) problems . ECP can be thought as an extension of Kelley's
cutting plane method, which uses iterative Newton's method to refine feasible area and ultimately solve a problem
Outer approximation is a basic approach for solving Mixed Integer Nonlinear Programming (MINLP) models
suggested by Duran and Grossmann (1986) . Based on principles of decomposition, outer-approximation and
relaxation, the proposed algorithm effectively exploits the structure of the original problems. The new problems
consist of solving an alternating finite sequence...
J.F. Benders devised an approach for exploiting the structure of mathematical programming problems with complicating
variables (variables which, when temporarily fixed, render the remaining optimization problem considerably more
tractable).The algorithm he proposed for finding the optimal value of this vector employs a cutting-plane approach for
building up adequate representations of...
The organization of general design problems into programming models allows for the defining and finding of their
(global) optimal solution. MINLP models represent problems as a sets of continuous variables with binary integer
variables. The continuous variables are restricted to defined constraints, and the binary variables represent whether
The Branch and Bound (BB or B&B) algorithm is first proposed by A. H. Land and A. G. Doig in 1960 for
discrete programming. It is a general algorithm for finding optimal solutions of various optimization problems,
especially in discrete and combinatorial optimization. A branch and bound algorithm consists of...
General disjunctive programming, GDP, is an alternative approach to represent the formulation of traditional
Mixed-Integer Nonlinear Programming, solving discrete/continuous optimization problems. By using algebraic
constraints, disjunctions and logic propositions, Boolean and continuous variables are involved in the GDP
formulation. The formulation process of GDP problem are more intuitive, and the...
The generalized disjunctive programming (GDP) was first introduced by Raman and Grossman (1994). The GDP extends
the use of (linear) disjunctive programming (Balas, 1985) into mixed-integer nonlinear programming (MINLP) problems,
and hence the name. The GDP enables programmers to solve the MINLP/MILP optimization problems by applying a
combination of algebraic...
Mixed-integer linear fractional programming (MILFP) is a category of mixed-integer linear programming (MILP). It is similar to MILP in that it uses the branch and bound
approach. It is widely used in process engineering for optimizing a wide variety of production processes ranging from petroleum refinery to polymerization processses and...
Sigmoid problems are a class of optimization problems with the objective of maximizing the sum of multiple
sigmoid functions. They are defined by their limits at negative and positive infinity. Similar to the unit step
function the function approaches 1 as it approaches infinity and approaches -1 as it approaches...
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...
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...
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...
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...
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
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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
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.
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
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...