1. BT8118 – Adv. Topics in Systems Biology
Prof. Eivind Almaas
Dept. of Biotechnology, NTNU

2. More than simplex…
Simplex algorithm is powerful and often among the fastest available
However:
it is not useful for non-linear programming
it is not fastest for very large problems
Remember:
 Simplex method ONLY selects among Basic Feasible Solutions
Basic variables are non-zero, Non-basic variables are all zero
Alternative: Interior Point Methods

3. Interior point methods
Idea: 1. Follow interior path towards optimum from starting point
2. AVOID the boundary
3. Many possible approaches exist to follow an “interior” path

4. Interior point methods:
Central path
Linear program:
Barrier function or auxiliary objective functions:
Slack variable
Q: What happens when μ is zero?

5. Barrier function Q: Draw this function … Answer: is strictly concave
Example: f(x) = x + log(5-x)

6. Barrier function and central path
Maximum of is defined as
Central path defined as:

7. Interior point algorithm
Start with large value for μ
Find approximate value for
Reduce value for μ and repeat 2.
Note: Resulting optimal point is the
analytic center of all optimal solutions!

8. MatLab examples
Solve using (A) simplex and (B) interior point algorithms
Example 1:
Example 2:
Q: What is analytic center of optimal solutions?
(x1,x2) = (1.5,2)

9. Interior point method, and equational form LP Linear program:
Barrier function or auxiliary objective functions:
Q: Explain why this barrier function?

10. Sensitivity analysis, part II
Shadow price:
Reduced cost:
PhPP defined using Shadow Prices

11. Complementary slackness
(D)
The complementary slackness condition is:
Or equivalently, with xs and ys as explicit slack variables

12. Complementary slackness:
Interpretation
Optimality can only be achieved when the condition of complementary slackness holds.
Consequently: conditions for optimality simply consists of
Primal feasibility
Dual feasibility
Complementary slackness
These conditions valid for optimality not only in LP, but also in non-linear programming! (the Karush-Kuhn-Tucker conditions)

13. Complementary slackness
Proof

14. Geometrical Interpretation
Linear program:
Primal Feasibility:
Dual Feasibility:
Complementary slackness:

15. Geometrical Interpretation…
Introduce dual slack variables:
New optimality conditions:
Primal feasibility: ………..
Dual feasibility: ………….
Complementary slackness:

16. Geometrical Interpretation…
We may re-write the dual feasibility condition in matrix form:
Note: The vectors in front of dual variables correspond to normal vectors of primal constraints!
Each dual variable corresponds to one constraint
Non-binding primal constraint  corresponding dual variable zero.
The optimality conditions  optimal solution at basic feasible solutions where pos. lin. comb. of normals may generate c.

17. Sensitivity analysis

18. Quick summary of course

19. FBA idea
Most used (and only currently realistic) method for modeling genome-scale metabolism
Based on:
List of all possible reactions
Mass conservation
Steady-state
Optimization of a cellular objective

20. Experiment: E. coli growth on glycerol
Study adaptive growth of E. coli on glycerol:
60-day experiment
Three independent populations:
E1 & T=30C; T=37C
Initially sub-optimal performance
glycerol
R.U. Ibarra, J.S. Edwards & B.O. Palsson, Nature 420, 186 (2002)

21. Principles of metabolic reconstruction
Thiele & Palsson, Nature Protcols, 5:95 (2010)

22. Linear programming
Basic feasible solutions == vertices in solution space
Simplex algorithm is method to move between basic feasible solutions
We use slack variables and auxiliary linear program to obtain equational form and starting point for the simplex algorithm.

23. Simplex algorithm

24. Hit-and-run method Algorithm:
Find random starting point in null space (red)
Pick random direction
Move in random direction until boundary is hit.
Specular reflection from boundary
After k reflections, randomize direction
Store coordinates every time distance d is traversed since last sample point.
Why not Monte Carlo sampling? …

25. Flux Variability Analysis

26. Minimization of Metabolic Adjustment (MoMA)

27. Bilevel optimization
Engineering
Biology

28. Duality theorem of LP

29. Assessment of sensitivity of objective function
Phenotype Phase-Plane (PhPP) Analysis
Central tool for PhPP:
Assessment of sensitivity of objective function
What is response of objective function Z to
Changes in internal fluxes?  reduced costs
Changes in the bounds vector?  shadow price

30. Interior point methods
Idea: 1. Follow interior path towards optimum from starting point
2. AVOID the boundary
3. Many possible approaches exist to follow an “interior” path

31. Complementary slackness:
Interpretation
Optimality can only be achieved when the condition of complementary slackness holds.
Consequently: conditions for optimality simply consists of
Primal feasibility
Dual feasibility
Complementary slackness
These conditions valid for optimality not only in LP, but also in non-linear programming! (the Karush-Kuhn-Tucker conditions)