最优化期末考试.doc

Definition: Convex combination Definition: Convex combination Theorem: Properties of convex setsDefinition: Convex functionsTheorem: Properties of convex functionsTheorem: First-order ConditionsTheorem: Second-order ConditionsTheorem: Second-order ConditionsQuestions: Definition: Convex programmingTheorem: Properties of convex programmingThe geometrical properties for linear programming The feasible region D for a LP is convex if D is nonempty. If there is a feasible solution, then there is at least one basic feasible solution. A feasible solution is a basic feasible one if and only if it is an extreme point of the feasible region D If a LP has optimal solutions, then at least one such solution is a basic feasible one. If a LP is feasible and bounded, then it has at least one optimal solution.？Show that the following model is convex programming ？Show that the following results ？Determine whether the following statements are true or not The linear programming is a kind of convex programming. If x* is the only optimal solution to a LP, then it is just an extreme point of the feasible region. For a LP, the optimal solution must be one of extreme points of the feasible region. For a LP, the optimal basic feasible solution must be one of extreme points of the feasible region. For a LP, if the feasible region is nonempty, then there certainly exist at least one optimal solution.对偶单纯形法The dual theorem and its consequencesTo simplify the exposition, we consider the following symmetrical dual problems*:*Remarks: for asymmetrical dual problems, the conclusions discussed are also applicableWeak DualityTheorem: The Dual TheoremTheorem: Complementary Slackness The Dual Simplex MethodTheorem： For the simplex tableau？Use the Dual Theorem to solve the following LP :Where it is known that the optimal sol