1. Solver & Optimization Problems
An optimization problem is a problem in which we wish to determine the best values for decision variables that will maximize or minimize a performance measure subject to a set of constraints
A feasible solution is set of values for the decision variables which satisfy all of the constraints
An optimal solution is a feasible solution which achieves the maximization/minimization objective for the performance measure
Solver is an Excel Add-in which can identify the optimal solutions for a correctly defined spreadsheet model

2. Components of an Optimization Problem
Decision Variables: Changing cells, the input parameters users experiment with to try to improve the situation and which are under the user’s control
Constraint Cells: The performance measures that users watch to make sure that cell values remain in an appropriate range
Objective: Set or Target cell, the key performance measure that the user wants to maximize or minimize

3. Solver Modeling Requirements
All components of the optimization problem must be on the same worksheet. Solver’s settings are saved with the sheet.
To speed up computation time, keep reports, data sets used to calculate parameter values, and other intermediate calculations on a different worksheet.
Solver’s constraint dialog box will not let you enter formulas. All formulas and calculations must be done on the worksheet. The constraint dialog box just compares cells to determine feasibility.

4. Overview of Mathematical Programming Optimization Techniques
Linear Programming:
Continuous values for decision variables
Linear constraints
Single linear objective
Nonlinear Programming:
Linear or nonlinear constraints
Single linear or nonlinear objective

5. Overview of Mathematical Programming Optimization Techniques (continued)
Integer Programming:
Integer values for decision variables
Linear constraints
Single linear objective

6. Integrality Considerations
In linear and non-linear programming, the decision variables are not required to assume only integer values. Therefore often fractional solutions are identified as the optimal solution.
If one or more decision variables need to consider only integer values, the model becomes an integer programming problem.
If possible, fractional solutions can be rounded, interpreted as the average number or work-in-progress or ignored if the model is for planning purposes only

7. Linearity
A linear function is where each variable appears in a separate term together with its constant coefficient.
The graph of a linear function of two variables is a straight line
An optimization problem is linear if:
the objective is a linear function of the decision variables
Each constraint cell is a linear function of the decision variables

8. Types of LP Solutions
Any linear program falls in one of three categories:
has a unique optimal solution or alternate optimal solutions
is infeasible (the problem is overconstrained so that no solution satisfies all the constraints)
has an objective function that can be increased without bound

9. Solver Result Messages
Solver found a solution. All constraints and optimality conditions are satisfied: Solver has correctly identified an optimal solution for the problem you have formulated. Note that there may be alternative optimal solutions possible however.
Solver has converged to the current solution. All constraints are satisfied: You have not selected the linear programming option in the Solver options. Thus nonlinear programming is being performed and this is the best solution Solver has found so far. It is not guaranteed to be the optimal one however.

10. Solver Result Messages
The Linearity Conditions required by this Solver Engine are not satisfied: Solver’s preliminary tests indicate that your model is not linear but you have asked Solver to do Linear Programming. If your model is not linear, you need to change the Solver Engine to the GRG Nonlinear engine instead of the Simplex LP engine.

11. Convex Set of Points
A set of points is convex if it has the following property:
Consider all possible pairs of points in the set and consider the line segment connecting any such pair. All such line segments must lie entirely within the set.

12. Convex –vs- Nonconvex

13. Highly Nonlinear Problems Highly Nonlinear Problems
General Convex Problems
Quadratic Problems
Linear Problems

14. Quadratic Programming
Linear convex constraints
Objective is a quadratic function:
Ai xi2 +  Bijxixj +  Cixi + D
Portfolio structuring uses quadratic programming models
Unique optimal solution exists that can be found using Solver’s generalized reduced gradient (GRG) algorithm

15. Solver Result Messages
Solver could not find a feasible solution: You may have too many constraints, one of the constraints may be entered wrong (e.g. the inequality sign might be going the wrong way) or you may not have enough changing cells.
Set Cell values do not converge: Your model as formulated is unbounded. One or more constraint is missing from the problem or entered wrong. Often times the modeler has forgotten to check the Assume Nonnegativity option in Solver.

16. Solver Result Messages
Solver encountered an error value in a target or constraint cell: Using the optimization technique selected, a cell formula resulted in an error message and the algorithm cannot continue solving the problem.
This can happen when your formula results in a number that is not real (for instance, when you divide by zero). You will need to fix the logic and then close down and reopen Excel to clear the registry of this error message.