Optimization Models
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1. Introduction
1.1. Overview
1.2. Optimization Models
2. Linear Algebra
2.1. Vectors
2.2. Matrices
2.3. Linear Equations
2.4.  Least-Squares
2.5. Eigenvalues
2.6  Singular Values
3. Convex Models
3.1. Convexity
3.2. LP and QP
3.3. SOCP
3.4. Robust LP
3.5. GP
3.6. SDP
3.7. Non-Convex Models
4. Duality
4.1. Weak Duality
4.2. Strong Duality
4.3.  Applications
5. Case Studies
5.1. Senate Voting
5.2. Antenna Arrays
5.3. Localization
5.4. Circuit Design

Matrices

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Matrices are collections of vectors of same size, organized in a rectangular array. The image shows the matrix of votes of the 2004-2006 US Senate. (Source.)

Via the matrix-vector product, we can interpret matrices as linear maps (vector-valued functions), which act from an ‘‘input’’ space to an ‘‘output’’ space, and preserve addition and scaling of the inputs. Linear maps arise everywhere in engineering, mostly via a process known as linearization (of non-linear maps). Matrix norms are then useful to measure how the map amplifies or decreases the norm of specific inputs.

We review a number of prominent classes of matrices. Orthogonal matrices are an important special case, as they generalize the notion of rotation in the ordinary two- or three-dimensional spaces: they preserve (Euclidean) norms and angles. The QR decomposition, which proves useful in solving linear equations and related problems, allows to decompose any matrix as a two-term product involving an orthogonal matrix and a triangular matrix.

Outline