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## week 10

$$\def\Real{\mathbb{R}} \def\Comp{\mathbb{C}} \def\Rat{\mathbb{Q}} \def\Field{\mathbb{F}} \def\Fun{\mathbf{Fun}} \def\e{\mathbf{e}} \def\f{\mathbf{f}} \def\bv{\mathbf{v}}$$

#### Simplex method

Tableau, pivot step, pivot rules. Exceptions. Phase one (finding a feasible point).

#### Sample problem

1. For which $$h$$ the following system is feasible?
$\begin{array}{ccc} &y&\leq 0\\ x&&\geq 0\\ x&-y&\leq ## mock midterm 2 $$\def\Real{\mathbb{R}} \def\Comp{\mathbb{C}} \def\Rat{\mathbb{Q}} \def\Field{\mathbb{F}}$$ #### Sample problems 1. Find Fenchel dual (conjugate, or $$\check{f}(p):=\sup_x px-f(x)$$) for 1. $$f(x)=|x-1|+x/2$$; Answer: \[ \check{f}(p)=p-1/2 \mathrm{\ if}\ -1/2\leq p\leq 3/2; +\infty \mathrm{\ otherwise}.$
2. f$$(x)=x^2/2+1/2 \ \mathrm{for} \ |x|\leq 1; |x| \ \mathrm{otherwise}$$.

$\check{f}(p)=p^2/2-1/2 ## week 9 $$\def\Real{\mathbb{R}} \def\Comp{\mathbb{C}} \def\Rat{\mathbb{Q}} \def\Field{\mathbb{F}} \def\Fun{\mathbf{Fun}} \def\e{\mathbf{e}} \def\f{\mathbf{f}} \def\bv{\mathbf{v}}$$ #### Linear Proramming Barvinok, 4. Convex closed cones, their polars. General LP problems. Dual problems for general LP. Polyhedral cones, strong duality. #### Sample problems 1. Let $$K=\Real_+^3$$ be the cone of vectors with ## week 8 $$\def\Real{\mathbb{R}} \def\Comp{\mathbb{C}} \def\Rat{\mathbb{Q}} \def\Field{\mathbb{F}} \def\Fun{\mathbf{Fun}} \def\e{\mathbf{e}} \def\f{\mathbf{f}} \def\bv{\mathbf{v}}$$ #### Convex optimization Lange, ch.5. Convex programs. Convexity of the set of optima. Constraint qualification: Slater condition. #### Duality Boyd-Vandenberghe, 5.1-3,5.6. Lagrange dual functions. Dual optimization problems. Fenchel duals (conjugate) functions. Weak duality … ## week 7 $$\def\Real{\mathbb{R}} \def\Comp{\mathbb{C}} \def\Rat{\mathbb{Q}} \def\Field{\mathbb{F}} \def\Fun{\mathbf{Fun}} \def\e{\mathbf{e}} \def\f{\mathbf{f}} \def\bv{\mathbf{v}}$$ #### Convexity Convex sets. Convex hulls. Supporting hyperplanes. Convex functions; properties. Log-convex functions. #### Sample problems 1. Is the set of continuous functions on $$\Real^2$$ positive at all points with rational coordinates numbers convex? ## weeks 5 and 6 $$\def\Real{\mathbb{R}} \def\Comp{\mathbb{C}} \def\Rat{\mathbb{Q}} \def\Field{\mathbb{F}} \def\Fun{\mathbf{Fun}} \def\e{\mathbf{e}} \def\f{\mathbf{f}} \def\bv{\mathbf{v}}$$ #### Quadratic forms Positive definite quadratic forms. Signatures, and relations between the indices of inertia of the (that is the number $$n_+$$ of positive, $$n_-$$ negative and $$n_0$$ ,zero coefficients in the … ## programming exercise 1. $$\def\Real{\mathbb{R}} \def\Comp{\mathbb{C}} \def\Rat{\mathbb{Q}} \def\Field{\mathbb{F}} \def\Fun{\mathbf{Fun}} \def\e{\mathbf{e}} \def\f{\mathbf{f}} \def\bv{\mathbf{v}}$$ Use any language or library. The assignments are due by midnight of 3.14 (Monday). #### 3 credits students Consider the piecewise linear function $$f$$ on $$[-4,4]$$ with nodes at the integer … ## mock midterm 1 $$\def\Real{\mathbb{R}} \def\Comp{\mathbb{C}} \def\Rat{\mathbb{Q}} \def\Field{\mathbb{F}} \def\Fun{\mathbf{Fun}} \def\e{\mathbf{e}} \def\f{\mathbf{f}} \def\bv{\mathbf{v}}$$ 1. Consider the following subsets of $$\Real^2$$: \[ A=\{(x,y):xy>0\}; B=\{(x,y):x=0\}; C=\{(x,y):y=0\}; D=\{(0,0)\}.$
Which of the following sets is open? closed?
$A; B; A\cup C; A\cup B\cup C; A\cup D$

## week 4

$$\def\Real{\mathbb{R}} \def\Comp{\mathbb{C}} \def\Rat{\mathbb{Q}} \def\Field{\mathbb{F}} \def\Fun{\mathbf{Fun}} \def\e{\mathbf{e}} \def\f{\mathbf{f}} \def\bv{\mathbf{v}}$$

### Differentials

Differentials. Slope functions.

Inverse function and implicit function theorems.

Matrix differentials

Sample problems:

• Find the differential of
$(x,y)\mapsto (x^2-y^2,2xy)$
• The function $$z(x,y)$$ is implicitly given by
\[

## week 3

$$\def\Real{\mathbb{R}} \def\Comp{\mathbb{C}} \def\Rat{\mathbb{Q}} \def\Field{\mathbb{F}} \def\Fun{\mathbf{Fun}} \def\e{\mathbf{e}} \def\f{\mathbf{f}} \def\bv{\mathbf{v}}$$

### Background in Linear Algebra and Analysis

Topology: closed and open sets. Convergence. Compact sets. Criterion of compactness in $$\Real^n$$.

Continuity.

Sample problems:

• Find an open and set in $$\Real$$