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Linear Programming
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When a problem asks you to formulate you need to provide your mathematical formulation of the LP with a clear justification of variables

INSTRUCTIONS TO CANDIDATES
ANSWER ALL QUESTIONS

This assignment consists of eight problems that require you to submit a written response and a coding component.

(1) When a problem asks you to formulate you need to provide your mathematical formulation of the LP with a clear justification of variables, constraints and objective function. You should submit your written response as a PDF to GradeScope and match the page number with the questions that you answered. You can find detailed instructions on how to scan and submit your assignments on Canvas. If you fail to match the page to the corresponding question, the marker will not be able to view your response, and thus you will be awarded a 0 mark for the question.

(2) When a problem involves computation you must give the Python source code that produces the result, the final numerical results and an interpretation of your numerical results. The Python code and numerical results go into a Jupyter Notebook file QBUS2310-2022S2-Assignment1-code.ipynb. You should download the file from Canvas and enter your code in the space provided. You should submit your code as a Jupyter notebook file via Canvas. In addition, the final numerical results and your interpretation of the results should appear in the PDF of your written response, described in (1).

Note: only problems 1 and 2 do not involve computation.

The assignment is due by Monday, the 5th of September, 5pm. Late assignments will not be accepted unless a special consideration was granted.

The problems have unequal weight. Some are easy. Others, not so much.

1. (10 points) Formulate the following problem as LP: Given A ∈ Rm×n, b ∈ Rm,mminimize Σ max{0, aT x + bi}. The variable is x ∈ Rn. i=1

 2. (10 points) In  linear  classification  problems  we  are  given  two  sets  of  points  in  Rn,  x1, . . . , xN and    y1, . . . , yM  , and wish to find an affine function f  : Rn R that is positive on the first set and negative on the second. When the two sets of points cannot be linearly separated, we might

seek an affine function that approximately classifies the points, for example, one that minimizes the number of points misclassified. Unfortunately, this is in general a difficult combinatorial optimization problem. One heuristic for approximate linear separation is based on support vector classifiers, which we discussed in class (see Topic 4: Piecewise-linear optimization).

We start with the basic feasibility problem:

aT xi + b ≥ 1, i = 1, . . . , N, aT yi + b ≤ −1, i = 1, . . . , M.

We then relax the constraints by introducing nonnegative variables u1, . . . , uN and v1, . . . , vM , and forming the inequalities

aT xi + b ≥ 1 − ui, i = 1, . . . , N, aT yi + b ≤ −(1 − vi), i = 1, . . . , M. (1)

When u = v = 0, we recover the original constraints. By making u and v large enough, these inequalities can always be made feasible. We can think of ui as a measure of how much the constraint aT xi    b     1 is violated, and similarly for vi. Our goal is to find a, b, and sparse nonnegative u and v that satisfy the inequalities (1). Formulate this problem as an LP.

3. We consider an illumination system of m lamps, at positions l1, . . . , lm R2, illuminating n flat patches.

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