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

Protein folding

Protein folding 

The protein folding problem is to predict the three-dimensional structure of a protein based on the sequence of the amino-acids that constitutes it. The amino-acids interact with each other (for example, they may be electrically charged). Such a problem is difficult to address experimentally, which calls for computer-aided methods.

In recent years, some authors have proposed to express the problem as an optimization problem, involving the minimization of a potential energy function, which is usually a sum of terms reflecting the interactions between pairs of amino-acids. The overall problem can be modeled as a non-linear optimization problem.

Protein folding: energy function 

Unfortunately, protein folding problems remain challenging. One of the reasons is the size of the problem (number of variables and constraints). Another difficulty comes from the fact that the potential energy function (which the actual protein is minimizing) is not exactly known. Finally, the fact that the energy function is usually not linear, and not even convex, may lead algorithms to discover ‘‘spurious’’ (that is, wrong) molecular conformations. The picture on the left is a three-dimensional rendition of the level sets of the energy function.