The optimal solution is the point that maximizes or minimizes the objective function, and the optimal value is the maximum or minimum value of the function. The context of a problem determines whether we want to know the objective function’s maximum or the minimum value. If a linear programming problem represents the amount of

In a convex optimization problemany local optimal solution is a global one Proof. Let x be a local optimum, i.e. there is R >0 s.t. f(x) f(z) 8z 2 \B(x;R): By contradiction, assume that x is not a global optimum, i.e., there is y 2 s.t. f(y) <f(x ). Take 2(0;1) s.t. x + (1 )y 2B(x ;R). Then we have f(x) f( x + (1 )y) f(x) + (1 )f(y) <f(x);

Describe where an optimal solution can be located in the feasible region of a linear programming problem. Identify the different possibilities for how many optimal solutions a linear programming problem can have. Explain how a linear programming problem could have no optimal solution.

Determine optimal mixture of ingredients that will minimize costs. Identify decision variables. Formulate the objective function.

Non-dominated solution set Given a set of solutions, the non-dominated solution set is a set of all the solutions that are not dominated by any member of the solution set The non-dominated set of the entire feasible decision space is called the Pareto-optimal set The boundary defined by the set of all point mapped

solves problems with one or more optimal solutions. The method is also self-initiating. It uses itself either to generate an appropriate feasible solution, as required, to start the method, or to show that the problem has no feasible solution. Each of these features will be discussed in this chapter.

1.5 Existence of Optimal Solutions Most of the topics of this course are concerned with • existence of optimal solutions, • characterization of optimal solutions, and • algorithms for computing optimal solutions. To illustrate the questions arising in the first topic, consider the following optimization problems: • 1+x (P) min x 2x s.t ...

optimal feasible solution, or simply an optimal solution. Definition: The optimal set is the set of optimal solutions; that is, the set of feasible solutions at which the objective function f takes on its optimal value (if it exists). Remark: In the above problem, the optimal value is 1 and the set of optimal solutions is the unit circle.

Solutions that satisfy the constraints are called feasible solutions. A feasible solution for which the optimization function has the best possible value is called optimal solution. Ex: Problem: Finding a minimum spanning tree from a weighted connected directed graph G.

What is the optimal solution? Give its objective function value and the value of all the decision variables. Solution. The initial feasible solution is xFG = xFG = xR = 0, s1 = 12, s2 = 14, s3 = 16. The optimal solution is xFP = 400, xFG = 0, xR = 100, s1 = 0, s2 = 0, s3 = 6, with objective function value 3200.

Our aim is to iterate toward an optimal solution, starting with this solution. A little bit of reflection should convince you that the present scenario is essentially the same as that at the start of Phase II of the standard Simplex method.

How to determine a starting basic feasible solution (BFS) for general LP? One technique is constructing a so-called Phase I Problem, and uses the Simplex Method itself to solve the Phase I LP problem for which a starting BFS is known, and for which an optimal basic solution is a BFS for the original LP problem if it’s feasible.

Compute the initial basic feasible solution and write its value for all of the problem’s variables (regardless of whether they are present in the original formulation or introduced for the canonical form).

Principle of Optimality: If b – c is the initial segment of the optimal path from b – f, then c – f is the terminal segment of this path. In practice: carry out backwards in time. Need to solve for all “successor” states first. Recursion needs solution for all possible next states. Doable for finite/discrete state-spaces (e.g., grids).

First question: How does one recognize or certify an optimal solution to a generally constrained and objectived optimization problem? Answer: Optimality Condition Theory.

To look for the best feasible solution, we can start from an arbitrary point, for example the vertex (0;0). We can then divide the plane into two regions: the set of points whose cost is greater than or equal to the cost of (0;0), that is the set of points such that x 1 + x 2 0, and the set of points of cost lower than the cost of (0;0), that is, 3

we demonstrate how to use Excel spreadsheet modeling and Solver to find the optimal solution of optimization problems. If the model has two variables, the graphical method can be used to solve the model.

Apply the simplex algorithm to compute an optimal solution. Always pivot in the column with largest reduced cost. Write down the optimal solution and optimal objective function value. Is the optimal solution that you found unique? Why? Solution. Pivot column: x

Common applications: Minimal cost, maximal profit, minimal error, optimal design, optimal management, variational principles. What to look for in setting up an optimization problem? What features are advantageous or disadvantageous? What devices/tricks of formulation are available? How can problems usefully be categorized?

Ellipsoid algorithm: It starts with an ellipsoid that includes the optimal solution, and keeps shrinking the ellipsoid until the optimal solution is found. This was the first poly-time algorithm, and was a theoretical breakthrough.