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

Control of a unit mass

Consider the problem if transferring a unit mass at rest sliding on a plane from a point to another at a unit distance. We can exert a constant force of magnitude x_i on the mass at time intervals i-1<t le i, i=1,ldots,10.

Denoting by y_1 the position at the final instant T=10, we can express via Newton's law the relationship between the force vector x and position/velocity vector y as y=Ax, where A in mathbf{R}^{2 times 10}.

Now assume that we would like to find the smallest-norm (in the Euclidean sense) force that puts the mass at y=(1,0) at the final time. This is the problem of finding the minimum-norm solution to the equation Ax = y. The solution is x_{rm LN} = A^T(AA^T)^{-1}y.