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  • 1

    Paula Amaral

    Lower bounds for quadratic fractional functions

    FCT Universidade Nova de Lisboa CMA (Centro de Matemática e Aplicações)

  • 2/48CVA2008 – FCT UNL Caparica

    • Fractional problems • Infeasible linear system • Global optimization method • Computational experience • Conclusions

    OUTLINE

  • 3/48CVA2008 – FCT UNL Caparica

    • Fractioal Quadratic Problem

    Fractional programming problems

    )( )( min

    Xx xg xf

    Stancu-Minasian (1999) – A fifth bibliography of fractional programming Schaible (1981) – Fractional Programming: applications and algorithms

    Engineering, business, finance, economics

    performance / cost income / investement cost / time

  • 4/48CVA2008 – FCT UNL Caparica

    • Fractioal Quadratic Problem

    Fractional programming problems

    Fractional programming : a tool for the assessment of sustainability Lara P. Stancu-Minasian I. (1999)

    Maximizing predictability in the stock and bond markets Lo A., Mackinlay C. (1997)

    Finantial planning with fractional goals Goedhart M., Spronk J. (1995)

    Discrete Fractional Programming techniques for location models Barros A. I. (1998)

  • 5/48CVA2008 – FCT UNL Caparica

    • Fractioal Quadratic Problem

    Linear/Linear

    Quadratic/Linear

    Quadratic/Quadratic

    Sum of ratios

    Charnes-Cooper (1962)

    Cambini (2002)

    Tuy, Konno (2004)

    Yamamoto, Konno (2007)

    )( )( min

    Xx xg xf

    linear , quadratic , concave-convex, polynomial fractional programs

  • 6/48CVA2008 – FCT UNL Caparica

    • Infeasible linear systems

    Production planning Product specifications Profits and costs Marketing Labour

    Production problem

    Infeasibility analysis

  • 7/48CVA2008 – FCT UNL Caparica

    mmibxa mibxa

    ii

    ii

    ,,1, ,,1,

    0

    0

    L

    L

    +== =≤

    update of old models

    integration of partial models

    unrealistic definitions.

    • Infeasible linear systems

  • 8/48CVA2008 – FCT UNL Caparica

    Remove constraints

    mmibxa mibxa

    ii

    ii

    ,,1, ,,1,

    0

    0

    L

    L

    +== =≤

    • Infeasible linear systems

  • 9/48CVA2008 – FCT UNL Caparica

    Find a solution of a close feasible system

    • Infeasible linear systems

  • 10/48CVA2008 – FCT UNL Caparica

    Ax (≤&=) b(A+H)x (≤&=) b+p

    Minimize Ψ(H,p) subject to

    x ∈X

    • Infeasible linear systems

  • 11/48CVA2008 – FCT UNL Caparica

    (A+H)x (≤&=) b+p

    Minimize Ψ(H,p) subject to

    x ∈X

    Ψ(H,p)=0 iff (H,p)=0

    Ψ(H,p) ≥0

    • Infeasible linear systems

  • 12/48CVA2008 – FCT UNL Caparica

    (A+H)x (≤&=) b+p x ∈ X

    Minimize Ψ(H,p)

    (H,p)=W

    Linear Programming Problem VATOLIN-2000

    Ψ(W)= Max |wij| i,j

    Ψ(W)= ΣΣ|wij| i,j

    Ψ(W)= Max Σ|wij|i j

    (l∞)

    (l1)

    (∞)

    • Infeasible linear systems

  • 13/48CVA2008 – FCT UNL Caparica

    (A+H)x (≤&=) b+p

    Minimize Ψ(H,p)

    (H,p)=W

    Ψ(W)= Max |wij|i,j

    Ψ(W)= ΣΣ|wij| i,j

    Ψ(W)= Max Σ|wij|i j

    (l∞)

    (l1)

    (∞)

    wij=±α for i=1,…,m, j=1,..n+1

    wij= ±αi j=k 0 j≠k

    • Infeasible linear systems

  • 14/48CVA2008 – FCT UNL Caparica

    • Infeasible linear systems

  • 15/48CVA2008 – FCT UNL Caparica

    (A+H)x (≤&=) b+p

    Minimize Ψ(H,p)

    (H,p)=W Ψ(W)= ΣΣ(wij)2 i,j

    (F)

    • Infeasible linear systems

  • 16/48CVA2008 – FCT UNL Caparica

    - 0.1365 - 0.1613 0.0522

    - 0.0714 - 0.0844 0.0273

    - 0.1065 - 0.1259 0.0407

    (H,p)=

    • Infeasible linear systems

  • 17/48CVA2008 – FCT UNL Caparica

    • Infeasible linear systems

  • 18/48CVA2008 – FCT UNL Caparica

    • Infeasible linear systems

    Total Least Squares Total Least Squares subset constraints

    Sabine van Huffel, The total least squares problem: computational aspects and analysis, Frontiers in Applied Mathematics, 9, SIAM, 1991.

    Amaral P, Barahona P. Connections between the total least squares and the correction of an infeasible system of linear inequalities. Linear Algebra and Applications 2005; 395: 191-210. Amaral P, Barahona P., A framework for optimal correction of inconsistent linear constraints. Constraints 2005; 10: 67-86.

    Amaral P, Júdice J, Sherali H D. A reformulation--linearization--convexification algorithm for optimal correction of an inconsistent system of linear constraints. Computers and Operations Research 2008; 35: 1494-1509.

  • 19/48CVA2008 – FCT UNL Caparica

    • Fractioal Quadratic Problem

    Linear/Linear

    Quadratic/Linear

    Quadratic/Quadratic

    Sum of ratios

    Aplications and caracterization

    Charnes-Cooper (1962)

    Cambini (2002)

    Tuy, Konno (2004)

    Yamamoto, Konno (2007)

  • 20/48CVA2008 – FCT UNL Caparica

    • Infeasible linear systems

    Quadratic/Quadratic

    Sum of Quadratic/Quadratic

  • 21/48CVA2008 – FCT UNL Caparica

    Why preserve the structure of zeros?

    • Infeasible linear systems

  • 22/48CVA2008 – FCT UNL Caparica

    • Infeasible linear systems

    hij=0 if aij=0

  • 23/48CVA2008 – FCT UNL Caparica

    • Infeasible linear systems

  • 24/48CVA2008 – FCT UNL Caparica

    • Infeasible linear systems

    KKT conditions

  • 25/48CVA2008 – FCT UNL Caparica

    • Infeasible linear systems

  • 26/48CVA2008 – FCT UNL Caparica

    • Infeasible linear systems

  • 27/48CVA2008 – FCT UNL Caparica

    • Infeasible linear systems

  • 28/48CVA2008 – FCT UNL Caparica

    • Infeasible linear systems

    WHY GLOBAL OPTIMIZATION ?

  • 29/48CVA2008 – FCT UNL Caparica

    • Infeasible linear systems

    SUM OF FRACTIONAL QUADRATIC FUNCTIONS with linear constraints

  • 30/48CVA2008 – FCT UNL Caparica

    Upper Bound

    Lower Bounds

    Optimal Value

  • 31/48CVA2008 – FCT UNL Caparica

    Branch & Bound

    Node poblems

    Branching Rule

    Upper bound

    Lower bound problem

    Node picking strategy

    • Global optimization method

  • 32/48CVA2008 – FCT UNL Caparica

    k

    k k

    Node problem

    • Global optimization method

    l2≤ x2≤u2 k k

    ln≤ xn≤un k k

    ls≤ xs≤us k k

    l1≤ x1≤u1 k k

    ls≤ xs≤us K+1 K+1

    ls≤ xs≤us K+2 K+2

  • 33/48CVA2008 – FCT UNL Caparica

    Lower bound

    • Global optimization method

  • 34/48CVA2008 – FCT UNL Caparica

    • Global optimization method

  • 35/48CVA2008 – FCT UNL Caparica

    l2≤ x2≤u2 k k

    ln≤ xn≤un k k

    ls≤ xs≤us k k

    l1≤ x1≤u1 k k

    ls≤ xs≤us K+1 K+1

    ls≤ xs≤us K+2 K+2

    Branching

  • 36/48CVA2008 – FCT UNL Caparica

    xs

    y=xs2

    lsk us k

    y=δskxs+βsk

    y

    xs*

    • Global optimization method

  • 37/48CVA2008 – FCT UNL Caparica

    xj

    y=xj2

    ljk=ljk+1 xj* ujk=ujk+2

    y=δjkxj+βjk

    y

    Branching Rule A and B

    • Global optimization method

  • 38/48CVA2008 – FCT UNL Caparica

    xs

    y=xj2

    ljk=ljk+1

    y=δjk+2xj+βjk+2

    ujk=ujk+2xs* =ujk+1 =ljk+2

    y=δjkxj+βjk

    y=δjk+1xj+βjk+1

    y

    Branching Rule A and B

    • Global optimization method

  • 39/48CVA2008 – FCT UNL Caparica

    Nodes inspection

    • Global optimization method

  • 40/48CVA2008 – FCT UNL Caparica

    Convergence

    • Global optimization method

  • 41/48CVA2008 – FCT UNL Caparica

    Pentium IV (Intel), CPU 3 GHZ, 2GB RAM, LINUX GAMS - MINOS

    Computational Experience

    • Computational experience

  • 42/48CVA2008 – FCT UNL Caparica

    { } 610

    ,1 −<

    − UBMax LBUB

    Nodes generated in the tree 1000

    Larger tolerances 10-ρ with 1 ≤ ρ ≤ 5

    li=0 ui=li+t with t=1, 5, 50, 500

    tolerances 10-ρ with 1 ≤ ρ ≤ 6

  • 43/48CVA2008 – FCT UNL Caparica

    Tolerances

    0

    1

    2

    3

    4

    5

    6

    7

    Ga len

    et Ite

    st2

    Ite st6

    Bg

    prt r

    Fo res

    t Wo

    od inf

    e Pr

    ob 4

    Pr ob

    5 Pr

    ob 6

    Pr ob

    7 Pr

    ob 8

    Pr ob

    9 Pr

    ob 10

    Pr

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