Simplex Algorithm in 1 Slide
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1 Administrivia 1
2 Canonical form: Simplex Algorithm in 1 Slide If we do pivot in A r,s >0, where c s <0 New cost value: New b vector: 2
3 The full Simplex Algorithm So far we have assumed that a basic feasible solution in canonical form is available to start the algorithm 3
4 The Simplex Algorithm Phase I Example Goal: We want to find a feasible solution Phase I: (i) Forget the cost function c T x. (ii) Introduce X 6,X 7 0. [One variable for each row] (iii) Solve 4
5 The Simplex Algorithm Phase I Theorem: (*2) has feasible optimal solution such that X 6 =X 7 =0 iff (*1) has feasible solution Remarks: (*2) is easy to convert to a feasible canonical solution (We will see) We can find its optimal solution (X 6 =X 7 =0) with the Phase II algorithm This is a feasible solution of (*1) 5
6 Tableaux Basic variable Objective (-w) X 1 X 2 X 3 X 4 X 5 X 6 X 7 RHS It s easy to convert this to a feasible canonical form: Basic variable Objective (-w) X 1 X 2 X 3 X 4 X 5 X 6 X 7 RHS -w X X
7 Tableaux Now, we can run the Phase 2 algorithm on this table to get a feasible solution of (*1): B. var Obj. (-w) X 1 X 2 X 3 X 4 X 5 X 6 X 7 RHS -w X X
8 Tableaux Now, we can run the Phase 2 algorithm on this table to get a feasible solution of (*1): B. var Obj. (-w) X 1 X 2 X 3 X 4 X 5 X 6 X -w X X RHS B. var Obj (-w) X 1 X 2 X 3 X 4 X 5 X 6 X 7 RHS -w /13*4-3+18/13* /13*3-2+18/13 18/ /13*7 X 3 0 4/13 2/13 1 3/13 1/13 1/ /13 X /13*4 1-5/13* /13*3 1-5/13-5/ /13*7 8
9 Tableaux B. var Obj (-w) X 1 X 2 X 3 X 4 X 5 X 6 X 7 RHS -w /13*4-3+18/13* /13*3-2+18/13 18/ /13*7 X 3 0 4/13 2/13 1 3/13 1/13 1/ /13 X /13*4 1-5/13* /13*3 1-5/13-5/ /13*7 Let us simplify this Table a little: B. var Obj (-w) X 1 X 2 X 3 X 4 X 5 X 6 X 7 RHS -w 1-7/13-3/13 0 2/13-8/13 18/13 0-6/13 X 3 0 4/13 2/13 1 3/13 1/13 1/ /13 X /13 3/13 0-2/13 8/13-5/13 1 6/13 9
10 Tableaux B. var Obj (-w) X 1 X 2 X 3 X 4 X 5 X 6 X 7 RHS -w X 3 0 3/8 1/8 1 1/4 0-1/8 0 5/4 X 5 0-7/8 3/8 0-1/4 1-5/8 1/8 3/4 All the relative costs are nonnegative ) optimal feasible solution. Phase I is finished. 10
11 Tableaux B. var Obj (-w) X 1 X 2 X 3 X 4 X 5 X 6 X 7 RHS -w X 3 0 3/8 1/8 1 1/4 0-1/8 0 5/4 X 5 0-7/8 3/8 0-1/4 1-5/8 1/8 3/4 B. var Obj (-z) X 1 X 2 X 3 X 4 X 5 X 6 X 7 RHS -z X 3 0 3/8 1/8 1 1/4 0-1/8 0 5/4 X 5 0-7/8 3/8 0-1/4 1-5/8 1/8 3/4 11
12 Tableaux B. var Obj (-z) X 1 X 2 X 3 X 4 X 5 X 6 X 7 RHS -z X 3 0 3/8 1/8 1 1/4 0-1/8 0 5/4 X 5 0-7/8 3/8 0-1/4 1-5/8 1/8 3/4 Make it to canonical form and continue with Phase II Do not make pivots in the column of X 6 and X 7 12
13 Simplex Algorithm with Matlab f = [ ]'; A = [ ]; b = [ ]'; lb = zeros(3,1); options = optimset('largescale','off','simplex','on'); [x,fval,exitflag,output,lambda] = linprog(f,a,b,[],[],lb,[],[],options); 13
14 Relevant Books Luenberger, David G. Linear and Nonlinear Programming. 2nd ed. Reading, MA: Addison Wesley, ISBN: Bertsimas, Dimitris, and John Tsitsiklis. Introduction to Linear Optimization. Athena Scientific Press, ISBN: Dantzig, Thapa: Linear Programming 14
15 Summary Linear programs: standard form, canonical form Solutions: Basic, Feasible, Optimal, Degenerate Simplex algorithm: Phase I Phase II Applications: Pattern classification 15
16 Convex Optimization CMU Convexity Part I Barnabás Póczos & Ryan Tibshirani
17 Goal of this lecture Review of Convex sets & Convex functions Definition Examples Basic properties Books to Read: Boyd and Vandenberghe: Convex Optimization, Chapters 2 & 3 Rockafellar: Convex Analysis 17
18 Line and Line Segments Definition [Line]: Definition [Line segment]: 18
19 Affine Sets Definition [Affine set]: Definition [Affine hull of set C]: Theorem [Affine hull]: The Aff[C] is the smallest affine set that contains C 19
20 Affine Sets Example Example [Solutions of linear equations]: The solution set of a system of linear equations is an affine set Solution set: Proof: 20
21 Boundaries Definition [x on boundary of C ( C) ]: Definition [x in interior of C]: Definition [relative interior (rel int C)]: 21
22 Boundaries Definition [closure of C (cl C) ]: Definition [relative boundary of C (rel C) ]: 22
23 Open and Closed Sets Definition [C closed]: Definition [C open]: Definition [C compact]: 23
24 Convex sets Convex Non convex Definition [Convex set]: Definition [Strictly convex set]: Example [Convex, but not strictly convex]: 24
25 Examples of Convex Sets empty set: singleton set: complete space: lines: line segments: hyperplanes: halfspaces: 25
26 Examples of Convex Sets Euclidian balls: L p balls, p 1 L p balls 0<p<1 26
27 Examples of Convex Sets Polyhedron: the solution set of a finite number of linear equalities and inequalities Matrix notation: Polytope: bounded polyhedron Intersection of halfspaces & hyperplanes 27
28 Convex hull Definition [Convex hull]: 28
29 Convex hull Convex hull properties: 29
30 Simplex: Examples of Convex Sets 30
31 Convex Combination Infinite many sums Integrals (Expected value) 31
![[Convex Cone]:](/docs-images/79/79813352/images/32-1.jpg "Definition [Conic")
32 Cones Definition [Cone]: Definition [Convex Cone]: Definition [Conic hull]: 32
33 Example: PSD matrices Definition [Positive semi definite matrix]: Theorem [eigenvalues]: A symmetric matrix A is positive definite iff all its eigenvalues are positive Partial ordering of square matrices: For arbitrary square matrices M, N we write M N if M N 0; i.e., M N is positive semi-definite. 33
34 Example: PSD matrices Theorem [Cone of PSD matrices]: The set of symmetric, PSD matrices form a convex cone: Proof: 34
35 Convex set representation with convex hull Theorem: [Representation of a closed convex set with a convex hull] 35
36 Dual representation Theorem: [Representation of a closed convex set with half spaces] 36
37 Convexity-preserving set operations Translation Scaling Intersection 37
38 Convexity-preserving set operations Affine function E.g. projection, dropping coordinates Set sum Direct sum 38
39 Convex Optimization CMU Convexity Part II Barnabás Póczos & Ryan Tibshirani
40 Conevex Hull Definition 1 Note: If C is a finite set, then this is closed polyhedron. If C contains infinite many points, then this can be open, closed, or none of them Theorem [Definition 2, Primal representation] A closed convex set is the intersection of all the closed half spaces containing S 40
41 Convex set representation with convex hull Theorem: [Representation of a closed convex set with a convex hull] Convex hull = convex combination of possibly infinite many points in the set.
42 Dual representation Theorem: [Representation of a closed convex set with half spaces] A closed convex set is the intersection of all the closed half spaces containing S
43 Convexity-preserving set operations Perspective projection (pinhole camera) 43
44 Convexity-preserving set operations Linear-fractional function (perspective function with affine function) Theorem: [Image of Linear fractional function] 44
45 Convexity-preserving set operations Application: [Conditional probabilities] 45
46 Convexity-preserving set operations Union doesn t preserve convexity 46
47 Separating hyperplane thm Theorem: [Separating hyperplane theorem] 47
48 Separating hyperplane thm Definition: [Strong separation] Definition: [Proper separation], Definition: [Strict separation] It "strictly separates" them if neither one touches the hyperplane. 48
49 Separating hyperplane thm Theorem: [Strong separation theorem] Counterexample: Why do we need at least one bounded set? 49
50 Separating hyperplane thm II Theorem: [Strong separation theorem II] 50
51 Supporting hyperplane thm Theorem: [Supporting hyperplane theorem] For any point x 0 on the boundary of convex C Theorem: [Partial converse of the supporting hyperplane theorem] 51
52 Use definition directly Proving a set convex Represent as convex hull Represent as the intersection of halfspaces Supporting hyperplane partial converse: C closed, nonempty interior, has supporting hyperplane at all boundary points C convex Build C up from simpler sets using convexity-preserving operations 52
53 Convex functions 53
54 Definition [convex function]: Convex functions Definition [strictly convex function]: 54
55 Definition [concave function]: Concave functions -f is convex 55
56 Geometric interpretation Convex functions 56
57 Strongly convexity Definition:[m-strongly convex function (m>0) An equivalent condition: Without gradient: With Hessian: A strongly convex function is also strictly convex, but not vice-versa. 57
58 Examples 58
59 Examples: Convex functions Convex Concave 59
60 Extended reals We can extend f from dom f to R n without changing its convexity Theorem: 60
61 Epigraph Definition [epigraph]: Theorem [convexity of the epigraph]: 61
62 Convex Function Properties 0 th order characterization This is useful, because we only need to check the convexity of 1D functions. Graph courtesy of Prof. Robert Freund 62
63 Convex Function Properties 1st order characterization The 1 st oder Taylor approximation is a global underestimator of f. Corollary: 63
64 Convex Function Properties 2nd order characterization Lemma 64
65 Jensen s inequality Theorem 65
66 Proving a function convex Use definition directly Prove that epigraph is convex via set methods 0 th, 1 st, 2 nd order convexity properties Construct f from simpler convex fns using convexity-preserving ops 66
67 Convexity-preserving fn ops Nonnegative weighted sum Pointwise max/sup Extension of pointwise max/sup 67
68 Convexity-preserving fn ops Affine map Composition Perspective map 68
69 Convex sets Representation: Summary convex hull, intersect hyperplanes supporting, separating hyperplanes operations that preserve convexity Convex functions epigraph 0 orders, 1st order, 2nd order conditions operations that preserve convexity 69
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