# lecture21 - CSE 6740 Lecture 21 How Do I Optimize With...

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CSE 6740 Lecture 21 How Do I Optimize With Constraints? (Constrained Optimization) Ravi Sastry and Alexander Gray agray@cc.gatech.edu Georgia Institute of Technology CSE 6740 Lecture 21 – p. 1/3 4

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Today 1. Convex (Constrained) Optimization Problems 2. Convex (Constrained) Optimization Methods 3. SMO: An Active Set Method CSE 6740 Lecture 21 – p. 2/3 4
Convex (Constrained) Optimization Problems Problems with a convex objective function and, if there are constraints, convex constraints. CSE 6740 Lecture 21 – p. 3/3 4

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Linear Programming When the objective and constraint functions are all affine, the problem is called a linear program (LP), which has the form Find x * = arg min x R D c T x + d (1) subject to Gx h (2) Ax = b. (3) CSE 6740 Lecture 21 – p. 4/3 4
Quadratic Programming When the objective function is quadratic and the constraint functions are affine, the problem is called a quadratic program (QP), which has the form Find x * = arg min x R D 1 2 x T Px + q T x + r (4) subject to Gx h (5) Ax = b. (6) CSE 6740 Lecture 21 – p. 5/3 4

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Quadratically Constrained QP If the constraints are also quadratic, the problem is called a quadratically constrained quadratic program (QCQP): Find x * = arg min x R D 1 2 x T Px + q T x + r (7) subject to 1 2 x T P i x + q T i x + r i 0 , i = 1 ,... ,M (8) Ax = b. (9) CSE 6740 Lecture 21 – p. 6/3 4
Second-order Cone Programming A closely related problem is called a second-order cone program (SOCP), which has the form Find x * = arg min x R D f T x (10) subject to || A i x + b i || 2 c T i x + d i , i = 1 ,... ,M (11) Fx = g. (12) A constraint of this form is called a second-order cone constraint. CSE 6740 Lecture 21 – p. 7/3 4

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Geometric Programming A geometric program (GP) is a problem of the form Find x * = arg min x R D f ( x ) (13) subject to c i ( x ) 1 , i = 1 ,... ,M (14) d i ( x ) = 1 , i = 1 ,... ,N (15) (16) where f and the c i have the form log p s k e a T 1 k x + b ik P (17) and the d i have the form e g T i x + h i . CSE 6740 Lecture 21 – p. 8/3 4
Semidefinite Programming A semidefinite program (SDP) has the form Find x * = arg min x R D c T x (18) subject to x 1 F 1 + ... + x n F n + G 0 (19) Ax = b. (20) CSE 6740 Lecture 21 – p. 9/3 4

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Relationships Generality: LP < QP < QCQP < SOCP < SDP. Computational cost: LP < QP < QCQP < SOCP < SDP. CSE 6740 Lecture 21 – p. 10/3 4
Optimization Methods The interior-point method. CSE 6740 Lecture 21 – p. 11/3

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## This note was uploaded on 04/03/2010 for the course CSE 6740 taught by Professor Staff during the Fall '08 term at Georgia Institute of Technology.

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lecture21 - CSE 6740 Lecture 21 How Do I Optimize With...

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