Second-Order Cone Program (SOCP) Detection and - Ampl

46 Pages·2010·0.45 MB·English

SOCP
Detectionand
Transforma-
tion
Second-Order Cone Program (SOCP) Detection
Erickson,
Fourer
and Transformation Algorithms for
Introduction
Optimization Software
Background
Detection
Transformation
Conclusion Jared Erickson [email protected]
Robert Fourer [email protected]
NorthwesternUniversity
INFORMSAnnualMeeting,Austin,TX
November 8, 2010
SOCP
Detectionand
Transforma-
tion
1 Introduction
Erickson,
Fourer
Introduction
2 Background
Background
Detection
Transformation 3 Detection
Conclusion
4 Transformation
5 Conclusion
Second-Order Cone Programs (SOCPs)
SOCP
Detectionand
Transforma-
tion
Can be written as a quadratic program
Erickson,
Fourer Not positive semi-definite
Introduction Convex
Background
Efficiently solvable with interior-point methods
Detection
Transformation SOCP general form:
Conclusion
minimize fTx
subject to (cid:107)A x +b (cid:107)2 ≤ (cTx +d )2 ∀ i
i i i i
cTx +d ≥ 0 ∀ i
i i
Introduction
SOCP
Detectionand
Transforma- Previous situation:
tion
Erickson, SOCPs can be written in numerous equivalent forms
Fourer
The form a modeler wants to use may not be the form a
Introduction
solver accepts
Background
Detection Converting the problem for a particular interior-point
Transformation solver is tedious and error-prone
Conclusion Ideal situation:
Write in modeler’s form in a general modeling language
Automatically transform to a standard quadratic
formulation
Transform as necessary for each SOCP solver
Motivating Example
SOCP
Detectionand
Transforma- (cid:113) (cid:113)
tion minimize (x +2)2+(y +1)2+ (x +y)2
Erickson,
Fourer Ampl model:
Introduction
Background var x;
Detection var y;
Transformation minimize objective:
Conclusion sqrt((x+2)^2+(y+1)^2)+sqrt((x+y)^2);
CPLEX 12.2.0.0: at2372.nl contains a nonlinear objective.
KNITRO 6.0.0: Current feasible solution estimate cannot be
improved.
objective 2.12251253;
30 iterations; 209 function evaluations
Motivating Example
SOCP
Detectionand
Transforma-
tion
Original:
Erickson,
Fourer
(cid:113) (cid:113)
Introduction minimize (x +2)2+(y +1)2+ (x +y)2
Background
Detection Transformed:
Transformation minimize u+v
Conclusion
(x +2)2+(y +1)2 ≤ u2
(x +y)2 ≤ v2
u,v ≥ 0
Motivating Example
SOCP
Detectionand
Transforma-
tion Ampl model:
Erickson,
Fourer
var x; var y;
Introduction
var u >= 0; var v >= 0;
Background
minimize obj: u+v;
Detection
s.t. C1: (x+2)^2+(y+1)^2 <= u^2;
Transformation
s.t. C2: (x+y)^2 <= v^2;
Conclusion
CPLEX 12.2.0.0: QP Hessian is not positive semi-definite.
KNITRO 6.0.0: Locally optimal solution.
objective 2.122027399;
3161 iterations; 3276 function evaluations
Motivating Example
SOCP
Original:
Detectionand
Transforma-
tion (cid:113) (cid:113)
Erickson, minimize (x +2)2+(y +1)2+ (x +y)2
Fourer
Introduction Transformed:
Background minimize u+v
Detection
r2+s2 ≤ u2
Transformation
Conclusion t2 ≤ v2
x +2 = r
y +1 = s
x +y = t
u,v ≥ 0
Motivating Example
SOCP
Detectionand Ampl model:
Transforma-
tion var x; var y;
Erickson, var u >= 0; var v >= 0;
Fourer
var r; var s; var t;
Introduction minimize obj: u+v;
Background s.t. C1: r^2+s^2 <= u^2;
Detection
s.t. C2: t^2 <= v^2;
Transformation
s.t. C3: x+2 = r;
Conclusion
s.t. C4: y+1 = s;
s.t. C5: x+y = t;
CPLEX 12.2.0.0: primal optimal; objective 2.121320344
5 barrier iterations
KNITRO 6.0.0: Locally optimal solution.
objective 2.122027305;
3087 iterations; 3088 function evaluations
Generally Accepted SOCP Form
SOCP
Detectionand
Transforma- SOCP general form:
tion
Erickson, minimize fTx
Fourer
Introduction subject to (cid:107)A x +b (cid:107) ≤ cTx +d ∀i
i i i i
Background
Detection where
Transformation x ∈ Rn is the variable
Conclusion
f ∈ Rn
Ai ∈ Rmi,n
bi ∈ Rmi
c ∈ Rn
i
d ∈ R
i