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The constrained-optimization problem (COP) is a significant generalization of the classic constraint-satisfaction problem (CSP) model. [1] COP is a CSP that includes an objective function to be optimized. Many algorithms are used to handle the optimization part.
Here some test functions are presented with the aim of giving an idea about the different situations that optimization algorithms have to face when coping with these kinds of problems. In the first part, some objective functions for single-objective optimization cases are presented.
A general chance constrained optimization problem can be formulated as follows: (,,) (,,) =, {(,,)}Here, is the objective function, represents the equality constraints, represents the inequality constraints, represents the state variables, represents the control variables, represents the uncertain parameters, and is the confidence level.
The equality constraint functions :, =, …,, are affine transformations, that is, of the form: () =, where is a vector and is a scalar. The feasible set C {\displaystyle C} of the optimization problem consists of all points x ∈ D {\displaystyle \mathbf {x} \in {\mathcal {D}}} satisfying the inequality and the equality constraints.
Consider the following nonlinear optimization problem in standard form: . minimize () subject to (),() =where is the optimization variable chosen from a convex subset of , is the objective or utility function, (=, …,) are the inequality constraint functions and (=, …,) are the equality constraint functions.
The optimization of portfolios is an example of multi-objective optimization in economics. Since the 1970s, economists have modeled dynamic decisions over time using control theory. [14] For example, dynamic search models are used to study labor-market behavior. [15] A crucial distinction is between deterministic and stochastic models. [16]
To see this, note that the two constraints x 1 (x 1 − 1) ≤ 0 and x 1 (x 1 − 1) ≥ 0 are equivalent to the constraint x 1 (x 1 − 1) = 0, which is in turn equivalent to the constraint x 1 ∈ {0, 1}. Hence, any 0–1 integer program (in which all variables have to be either 0 or 1) can be formulated as a quadratically constrained ...
In this example, the first line defines the function to be minimized (called the objective function, loss function, or cost function). The second and third lines define two constraints, the first of which is an inequality constraint and the second of which is an equality constraint.