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Download ISYE 6669: Deterministic Optimization - Course Guide & Syllabus and more Exams Operational Research in PDF only on Docsity!
Page 1 of 14 ISYE 6669 / ISYE6669 MIDTERM & FINAL EXAMS (LATEST 2025 / 2026 UPDATES STUDY BUNDLE PACKAGE WITH SOLUTIONS) DETERMINISTIC OPTIMIZATION UESTIONS & ANSWERS | 1009 CORRECT (VERIFIED ANSWERS) Linear Program ....ANSWER.....Functions all linear with continuous variable domain Nonlinear Program ....ANSWER.....Some linear dunctions with continuous variable domain Integer Program or Discrete Optimization ....ANSWER.....Linear and Nonlinear functions with discrete variable domain Unconstrained Optimization ....ANSWER.....A subclass of NLP with no constraints or simple bound constraints on the variables Page 2 of 14 Quadratic Program .....ANSWER.....A subclass of NLP where Objectives and constraints involve quadratic functions Mixed Integer Linear Program (MILP) .....ANSWER.....All linear functions where Some variables are continuous and some are discrete Mixed Integer Nonlinear Program (MINLP) .....ANSWER.....Some nonlinear functions where Some variables are continuous and some are discrete Mixed Integer Quadratic Program (MIQLP) sone ANSWER.....Nonlinear functions are quadratic where Some variables are continuous and some are discrete Page 4 of 14 Bounded Set .....ANSWER.....A set is bounded if it can be enclosed in a large enough (hyper)-sphere or a box Compact Set .....ANSWER.....closed and bounded set Convex Function ....ANSWER.....Function value at the average <= average of function values Linear function is convex and concave First Order Condition ....ANSWER.....first order Taylor's approximation is a global under-estimator Second Order Condition ....ANSWER.....Univariate function is convex if f"(x) >=0 If Hessian matrix is positive semi-definite Positive Semidefinite ....ANSWER.....A is psd if all its eigenvalues are nonnegative Page 5 of 14 x4TAx >= 0 for all x Operations preserving convexity ....ANSWER.....Nonnegative weighted sum of convex functions Maximum of convex functions Composition of convex functions if either f is nondecreasing or if each gi is a linear function Convex Set .....ANSWER.....a set in which every segment that connects points of the set lies entirely in the set Convexity Preserving Set Operations ....ANSWER.....Intersection, summation, Product Convex Optimization Problem ....ANSWER.....An optimization problem (in minimization) form is a convex optimization problem, Page 7 of 14 Unbounded Problem ....ANSWER.....there are feasible solutions with arbitrarily small objective values. Unbounded problem must be feasible Optimal Solution .....ANSWER.....If feasible solution x* is f(x*) <= f(x) for all x in X Weierstrass's Theorem .....ANSWER.....If f is a continuous function and X is a nonempty, closed and bounded set, then (P) has an optimal solution Global Solutions .....ANSWER.....lf P is a convex optimization problem, then a local optimal solution is a global optimal silution Problem Relaxation ....ANSWER.....A problem, g(x), if a relaxation of a function, f(x), if the feasible set is a subset of the original problem and f(x) >= g(x) for all x in X Optimal value of the relaxation provides a lower bound on the Page 8 of 14 original problem If the relaxation is infeasible then clearly the original problem is also infeasible Lagrangian Relaxation ....ANSWER.....Reread Module 6 Lesson 2 Gradient Descent .....ANSWER.....Reread Module 7 Lesson 2 Newton's Method ....ANSWER.....Reread Module 7 Lesson 3 Transportation Problem ....ANSWER.....There are m suppliers, n customers. Supplier i can supply up to si units of supply, and customer j has dj units of demand. It costs cij to transport a unit of product from supplier i to customer j. We want to find a transportation schedule to satisfy all the demand within minimum transportation cost Maximum Flow Problem ....ANSWER.....How much supply Bs can be transported from source Page 10 of 14 Line .....ANSWER.....A line consists of two rays starting at a point pointing two opposite directions Plane .... ANSWER.....a plane in R‘n is the set of solutions of one linear equation in n variables: a1x1 + +--+ anxn =c Halfspace .... ANSWER.....Half the whole space represented by: alx1] +++ anxn <=C alx] + +++ anxn >= C€ Polyhedron ....ANSWER.....A polyhedron is the intersection of a finite number of halfspaces Corner points are responsible for creating the set Convex Combination of Two Points ....ANSWER.....Given two points a, b € R4n, a convex combination of a, D is given by x = da +(1—A)b Page 11 of 14 Extreme Point ....ANSWER.....A point X in a polyhedron P is an extreme point if and only if x is not a convex combination of other two different points in P. Extreme points are "corner points". Convex Hull ....ANSWER.....A convex hull of m points @1,..., am is the set of all convex combinations of @1,..., am A nonempty and bounded polyhedron, polytope, is the convex hull of its extreme points Unbounded Polyhedron .....ANSWER.....is unbounded iff there are directions to move to infinity Page 13 of 14 If a polyhedron is unbounded, it may not have an extreme point Weyl-Caratheodory Theorem .....ANSWER.....Any nonempty polyhedron P with at least one extreme point can be formed by its extreme points X1,....x%m and its extreme rays @1,...,ek as follows P=conv{x1,....xm }+conic{e1,...,.ek} Active constraints ....ANSWER.....A linear constraint that is satisfied as equality at a given point is said to be active or binding at that point. Otherwise, if an inequality constraint is satisfied as strict inequality at a point, it is called inactive Linear independent constraints ....ANSWER.....If the normal directions of two or more linear constraints are linearly Page 14 of 14 independent, then these constraints are called linearly independent Basic solution .....ANSWER.....The unique solution of 7 linearly independent active constraints in R4n Basic feasible solution .....ANSWER.....A feasible basic solution which is an algebraic description of an extreme point Standard Form LP ....ANSWER.....Must have equality constraints and nonnegative constraints on all variables We assume the equality constraints are linearly independent To convert look at module 14 lesson 2 slide 4 
