1. Mathematical Programming
Mathematical models
Operations Research and Management (Decision) Science

2. Mathematical models
A mathematical model is a description of a system or problem using mathematical concepts, tools and language.
Mathematical model is a function, an equation, inequations, or system of equations or inequations, which represents certain aspects of the physical system or problem modelled.
Ideally, by the application of the appropriate techniques the solution obtained from the model should also be the solution to the system problem.

3. Optimization model Problem III
Suppose that a farmer has 35 acres of farm land to be planted with either cauliflower or kohlrabi. The farmer wants to have at least 8 acres of cauliflower. Farmer supposes increasing variable costs of cauliflower according to the function C1 = 0,25 x12-3x1, and costs of kohlrabi according to the function C2 = x22-4x2 (x1,2 – areas of plants).

4. Terminology Variables
Decision variables
Slack variables
Artificial variables
Constraints also called conditions or restrictions
Capacities or Capacity constraints
Requirements or Requirement constraints
Balance constraints
Definitional constraints
Objective function also called criteria function

5. Terminology Feasibility region Search space Choice set
Set of candidate solutions or
Set of feasible solutions
Objective function also called criteria function, cost function, energy function, or energy functional

6. Terminology Feasible solution Basic solution Infeasible solution
Optimal solution
Alternative solution
Suboptimal solution

7. Solution of problem III
x1 - area of cauliflower (ar)
x2 - area of kohlrabi (ar)
Minimize:
f(x) = 0,25x12 - 3x1 + x22 – 4x2  min (minimize the costs)
Subject to:
x1 + x2  35 (limit on total area)
x1  8 (limit on area of cauliflower )
x1 ≥ 0, x2 ≥ 0 (nonnegative area).

8. Mathematical programming
Optimization model
min f(x)  qi(x)  0 , i = 1, ..., m ,
xT=(x1, x2, ..., xn)T  Rn 
f(x) and qi (x) - real function of many variables and x – vector of variables from vector space Rn.

9. General optimality problems
Feasibility problem
The satisfiability problem, also called the feasibility problem, is just the problem of finding any feasible solution at all without regard to objective value.
Minimum and maximum value of a function
The problem of finding extrema of function without regard to some constraints.

10. General optimality problems
Mathematical optimization (alternatively, optimization or mathematical programming) refers to the selection of a best element from some set of available alternatives.
One constraints in the form of line (contour line, curve)
More then one constraints (subset of space of decision variables)

11. Classification of optimization models
Number of criteria
Single optimization
Multiple optimization
Type of criteria
Minimization
Maximization
Goal problem
Type of functions
Linear model
Nonlinear model
Convex model
Nonconvex model

12. Nonconvex or nonconcave function

13. Nonconvex feasibility region

14. Historical notes
Fermat and Lagrange found calculus-based formulas for identifying optima.
Newton and Gauss proposed iterative methods for moving towards an optimum.
Historically, the first term for optimization was "linear programming", which was due to George B. Dantzig, although much of the theory had been introduced by Leonid Kantorovich in 1939.
Dantzig published the Simplex algorithm in 1947
John von Neumann developed the theory of duality in the same year.