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Optimality and Efficiency- A Dual Objective Function for Optimization Algorithms

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**1. **Optimality and Efficiency- A Dual Objective Function for Optimization Algorithms C. L. Liu
National Tsing Hua University
Hsinchu, Taiwan

**2. **Optimization Problems

**3. **Constrained Optimization Problems

**4. **Solution of Optimization Problems Finding a Best possible solution at the
Lowest possible cost

**5. **Solution of Optimization Problems Cost : Time
Space
Energy
.
.
.
.
.

**6. **Solution of Optimization Problems Finding a Best possible solution at a
Lowest possible cost

**7. **Solution of Optimization Problems Finding a Best possible solution at a
Low cost
(Low : Polynomial in problem size)

**8. **Solution of Optimization Problems Finding a Best possible solution at a
High cost
(High : Exponential in problem size)

**9. **Solution of Optimization Problems Finding an Acceptable solution at a Low
cost

**10. **Approximation Algorithms (Performance Guaranteed)

**11. **Approximation Algorithms (Performance Guaranteed)

**12. **Next Fit Packing Algorithm

**13. **Approximation Algorithms (Performance Guaranteed)

**14. **Solution of Optimization Problems Finding an Acceptable solution at a tolerably
High cost

**15. **Simulated Annealing (Downhill - Uphill) Non-zero probability for up-hill moves
Probability depends on the magnitude of
up-hill move
Probability depends on total search time

**16. **Genetic Algorithm Start with an initial generation of solutions ( a finite
number, n, of randomly chosen solutions)
Crossover Operator : Combine two current solutions
to yield one or more offsprings
Mutation Operator : Mutate a current solution to
produce an offspring
Select from the pool the next generation of n
solutions according to a fitness criterion
Stop according to some stopping criteria : number
of generations, incremental improvement of the
best solution

**17. **An Example An optimal solution
An efficient algorithm
A special case of a well-studied problem

**18. **Linear Programming

**19. **Linear Programming

**20. **Linear Programming A consistent system (non-empty solution
space)

**21. **Linear Programming A redundant constraint
ai1 x1 + ai2x2 + ……+ aimxm ? bi

**22. **Linear Programming Remove a largest possible subset of
redundant constraints

**23. **Linear Programming
A smallest possible set of constraints

**24. **Graph Constraints Consistency
A redundant constraint
Removal of a largest possible subset of
constraints
A smallest possible subset of graph
constraints

**25. **Graph Constraints and Constraint Graph Graph Constraint
Constraint graph of a system of graph
constraints

**26. **Consistency Theorem
A system of graph constraints is consistent if and
only if its constraint graph has no directed positive
cycle

**27. **Redundancy An edge (u, v) is redundant if there is a
(directed) path from u to v, P, such that
w(u,v) ? w(P) and (u,v) is not on P

**28. **Redundancy

**29. **Redundancy

**30. **Removal of Redundant Constraints Theorem :
The Problem of removing a largest possible
subset of redundant constraints is NP-
complete
Proof :
Reduction from the directed Hamiltonian
Cycle Problem

**31. **Optimal Reduction

**32. **Zero Cycles Theorem
If G contains no zero-cycle, G – RG is the unique
optimal reduction of G where RG is the set of all
redundant edges of G

**33. **Zero Cycles

**34. **Optimal Reduction G1 and G2 are constraint graphs with the
same vertex set
WG1* (u, v) is the path length of the longest
path from u to v in G1
WG2* (u, v) is the path length of the longest
path from u to v in G2
Theorem :
G1 and G2 define the same solution space
if and only if WG1* (u, v) = WG2* (u, v) for
all u and v

**35. **Optimal Reduction For a given G, find G0 such that
(i) WG* (u, v) = WGo* (u, v) for all u and v
(ii) G0 has the minimum number of edges

**36. **Constraint Graph

**37. **Dependency Relation u and v are dependent if
(i) u = v , or
(ii) there is a zero closed walk that contains u and v
u and v are dependent if and only if
wG* (u, v) + wG* (v, u) = 0

**38. **Dependency Relation

**39. **Step 1

**40. **Step 2

**41. **Step 3

**42. **Step 4

**43. **Step 5

**44. **Why Minimum ?

**45. **Extensions axi ? bxj ? c
xi ? xj = xk ? xl
xi ? xj ? xk ? c

**46. **Final Remark