$ \vec x $ is the linear part of the objective, and the $ \vec x $ T $ \varrho \vec x $ is the quadratic part of the objective,which is specified by the $ \, n \times \, n $ matrix $ \varrho \times \vec x $ T denotes the vector transpose of the decision variables.The constraints are the same as they were in the LP formulation, and differ here only by the fact that they appear here in matrix notation. The objective and constraint left-hand sides are written here in matrix notation. This means that you always be assured of finding the absolute optimum solution.Ī quadratic program has the following standard formulation: Solutions to linear problems always represent a global maximum or minimum. Linear programs generally solve quickly, although large models can be computationally complex. In this standard form, all decision variables, x i, are real-valued and unconstrained, ranging from -INF to +INF (-∞ to ∞). This section provides more specific mathematical descriptions for each type of optimization.Ī linear optimization problem has the following standard formulation. The general description of optimization problems applies to all problem types. You can often achieve that with a simple transformation, combination, or disaggregation of the decision variables. Since linear and quadratic formulations are usually so much faster and easier to solve than nonlinear problems, it is worth careful thought to see if it is possible to reformulate a nonlinear problem so that it is linear or quadratic. There are often several ways to formulate the same problem. While the Frontline Solver engine used in the Analytica Optimizer is one of the better general-purpose optimization engines available, success with hard optimization problems depends on your ability to formulate the problem effectively, provide appropriate hints for the Optimizer, and adjust the search control settings. Many of the most famous and notoriously difficult computation problems can be cast as optimization programs, from the traveling salesman to the solution (or non-solution) of Fermat’s last “theorem.” It is, therefore, unreasonable to expect the Optimizer engine to succeed on any possible nonlinear problem you can formulate. But general nonlinear optimization is a computationally difficult problem. Linear and convex quadratic optimization problems are often relatively fast to compute. The problem is a nonlinear program (NLP) if the objective or any of the constraints are non-quadratic in any of the decision variables. For a quadratically-constrained program (QCP), the objective and constraints must all be linear or quadratic functions of decision variables. For a quadratic program (QP), the objective is a quadratic function and the constraints must be linear functions of the decision variables. But, you will find it helpful to have a basic understanding of what these classes mean.įor a linear program (LP), the objective must be a linear function of the decision variables. Analytica can automatically determine the type from analyzing the model, and choose the most suitable solver engine from those it has available. They define which solutions are feasible.Īn optimization problem may be linear, quadratic, or (other) nonlinear. g 1( $ \vec x $) ≤ b 1, are bounds on functions of the decision variables. If not specified, the lower and upper bounds are - INF and + INF - that is, there are no bounds. Each decision variable can have a lower bound and/or an upper bound. These bounds define the set of possible solutions, called the search space. For a nonlinear program (NLP), the objective can be any expression or variable that depends on the decision variables.Ī range lb i ≤ x i ≤ ub i, i = 1.n on the decision variables, defining what values are allowed. For a linear program (LP), the objective is defined by a set of coefficients or weights that apply to the decision variables. It gives a single number evaluating a solution, which the Optimizer tries to minimize or maximize, whichever you specify in the formulation. The Objective is a function, $ f(\vec x) $, of the decision variables. A solution is a set of values assigned to these decision variables. The formulation looks like this.Ī vector (one-dimensional array) of the decision variables whose values we can change to find an optimal solution $ \vec x $ = ( x 1, x 2, …, x n) n. An optimization problem is defined by four parts: a set of decision variables, an objective function, bounds on the decision variables, and constraints. The first step in performing an optimization is to formulate the problem appropriately. 4 Continuous, integer, and mixed-integer programs.3.4 Parts of a Non-Linear Program (NLP).3.3 Parts of a Quadratically-Constrained Program (QCP).3 Specific Optimization Characteristics.
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