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 c
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 r
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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
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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
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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
pro
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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
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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
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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 plan
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 (ahray
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
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
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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
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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
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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
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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
theo
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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
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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, epigra
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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 muc
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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 importan
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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
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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
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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 w
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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
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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
devel
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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 deali