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

Why do we take the log in GP?

For a posynomial f with values

f(x) = sum_{k=1}^K c_k x^{a_k},

where c >0, we have

log f(x) = log left( sum_{k=1}^K e^{a_k^Ty + b_k} right) = mbox{rm lse}(Ay + b),

where b_k := log c_k, k=1,ldots,K, and A is the K times n matrix with rows a_1,ldots,a_K.

We may ask, why take the logarithm of f? The function with values

g(y) = sum_{k=1}^K e^{a_k^Ty + b_k}

is convex already.

The answer is simply that the function g might take huge values, which might raise numerical problems. Taking its logarithm allows to recover smaller values.

Note that the logarithm is concave, while g is convex. A priori, log g is not convex. It turns out that ‘‘the exponential beats the logarithm’’, in the sense that the latter functions’ concavity is not enough to destroy the convexity of the exponential.