Homework #1, ISE 5405 - Optimization I, Fall 2015
Due September 8
Please turn in your typed solutions to this homework at the end of class on the due date and be
prepared to present your solution in class on that day. You may work with other students to d
ONE: INTRODUCTION
Linear programming is concerned with the optimization (minimization or
maximization) of a linear function while satisfying a set of linear equality and/or
inequality constraints or restrictions. The linear programming problem was first
c
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Topic #14: Fritz John Conditions
BSS Textbook Reading: Section 4.2
We now begin to develop optimality conditions for constrained optimization problems. We
begin by developing expressions for the set of improving directions and the set of feasible
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Topic #16: Constraint Qualifications
BSS Textbook Reading: Section 4.2, Chapter 5
We now concentrate on the topic of constraint qualifications (CQs), which are presented in
the following form.
If is a local minimum and some CQ holds , then is a K
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Topic #18: Quadratic Programming & Linear Complementarity Problems
BSS Textbook Reading: Section 11.1
We now demonstrate how we can apply the KKT conditions to solve a particular class of
problems, namely quadratic programming problems. We also d
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Topic #17: Optimality Conditions for Problems with Equality Constraints
BSS Textbook Reading: Section 4.3
Thus far, we have derived the Fritz John and KKT conditions for problems that contain only
inequality constraints. We now extend those conce
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Topic #19: Intro to Lagrangian Duality
BSS Textbook Reading: Sections 6.1 6.3
We now give a brief introduction to the topic of Lagrangian Duality (presented in detail in
Chapter 6 of the BSS textbook). Lagrangian duality is a powerful optimizatio
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Topic #20: Solving the Lagrangian Dual
BSS Textbook Reading: Sections 6.3 6.5
We now give an overview of two approaches for solving the Lagrangian dual. We begin by
examining the cutting plane algorithm, and we conclude by briefly discussing the
ISE 5405 - Optimization I, Fall 2015
Instructor: Douglas R. Bish, 207 Durham Hall ([email protected])
Oce Hours: TR 1:00-2:00 pm by appointment.
Teaching Assistant: Hrayer Aprahamian, 223 Durham Hall ([email protected])
Oce Hours: TBD.
Text: Linear Programming and
ISE-5405 - Optimization I, ICE #1 (solutions)
1. LP Inc. produces two chemicals, A and B, via two processes. Process 1 requires 2 hours of labor
and 1 lb of material X to produce 2 oz of A and 1 oz of B. Process 2 requires 3 hours of labor and
2 lb of mat
ISE-5405 - Optimization I, Fall 2015
ICE #1
Names:
1. LP Inc. produces two chemicals, A and B, via two processes. Process 1 requires 2 hours of labor
and 1 lb of material X to produce 2 oz of A and 1 oz of B. Process 2 requires 3 hours of labor and
2 lb o
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Topic #15: Intro to KKT Conditions
BSS Textbook Reading: Section 4.2
We now present the KKT conditions for inequality constrained problems. KKT points are a
subset of Fritz John points where the Lagrange multiplier for the objective gradient is
r
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Topic #12: Unconstrained Optimization
BSS Textbook Reading: Section 4.1
We now discuss some classical optimization results for unconstrained problems.
Univariate Functions
Let
. Then we know that is a local minimum
Since the domain is , we can de
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Topic #13: Theorems of the Alternative
BSS Textbook Reading: Section 2.4
We now turn our attention to theorems of the alternative, including Farkas Lemma and
Gordans Lemma. These results will be used in developing the Fritz John and KKT
optimalit
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Topic #3: Weierstrass, Closest Points, Separations, and Supports
BSS Textbook Reading: Sections 2.3 2.4
We now continue our discussion of preliminary concepts by discussing two important
theorems in optimization, followed by the concepts of separ
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Topic #2: Definitions and Background Material
BSS Textbook Reading: Sections 2.1 2.2
The goal of nonlinear programming is to obtain a globally optimal solution. However, as
we have seen, some solution algorithms will terminate with only a locally
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Topic #4: Introduction to Convex Functions
BSS Textbook Reading: Sections 3.1 3.2
We now discuss the convexity of functions in greater detail, including concepts related to
level sets, epigraphs, and the connection between convex sets and convex
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Topic #6: First Order Characterizations of Convexity
BSS Textbook Reading: Sections 3.1, 3.3
As discussed previously, the line between easy and difficult optimization problems does
not so much lie between linear and nonlinear programming problems
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Topic #5: Taylor Series and the Mean Value Theorem
BSS Textbook Reading: Section 3.3
We now discuss the concepts of Taylor series expansions and the mean value theorem,
which will be important in helping us characterize convex functions.
Differen
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Topic #7: Second Order Characterizations of Convexity
BSS Textbook Reading: Section 3.3
We continue to discuss the different ways in which we can show that a function is convex.
In particular, we now discuss a very popular way to demonstrate the
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Topic #8: Definiteness Properties for Matrices
BSS Textbook Reading: Section 3.3
In the last topic, we covered new ways to classify functions as convex or concave based
upon the definiteness properties of the Hessian matrix. We now discuss some c
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Topic #11: Generalized Convexity
BSS Textbook Reading: Section 3.5
We now discuss optimization results for minimizing functions that are convex,
quasiconvex, or pseudoconvex. Throughout, we will assume that
, where
is a non-empty convex set
Case
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Topic #9: Subgradients and Ascent/Descent Directions
BSS Textbook Reading: Sections 3.2, 3.5
We now generalize the results we developed for convex functions that are differentiable to
the non-differentiable case. We then discuss the concepts of a
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Topic #10: Optimization Results for Convex Programming Problems
BSS Textbook Reading: Section 3.5
We now discuss finding the maxima and minima of a convex function over a convex set. We
develop optimality conditions for a minimum and characterize
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Topic #1: Introduction to Nonlinear Programming
BSS Textbook Reading: Chapter 1
Throughout the semester, we will be dealing with nonlinear programming problems.
Specifically, we will be dealing with problems of the following form.
Minimize
()
Gen