A Polyhedral Approach to Cardinality Constrained Optimization
Ismael Regis de Farias Jr. and Ming ZhaoUniversity at Buffalo, SUNY
Summary
• Problem definition
• Relation to previous work
• Simple bound inequalities
• Further research
Problem Definition
Given c1n , Amn , bm1 , and ln1 , un1 ≥ 0, find xn1 that:
maximizes
cx
subject to
Ax b,
−l ≤ x ≤ u,
and at most k variables are nonzero
Motivation
• Portfolio selection
• Feature selection in data mining
Polyhedral approach
Derive within a branch-and-cut scheme strong
inequalities valid for:
Pi = conv {x Rn : jN aij xj bi , −l ≤ x ≤ u,
and at most k variables are nonzero},
i {1, …, m}, to use as cutting planes in the
branch-and-cut
Previous work
• Bienstock (1996): critical set inequalities
• de Farias and Nemhauser (2003): cover inequalities.
However, the present case is more general and
the polyhedral structure is much richer …
Example
Let P = conv {x [−1, 1]6 : 6 x1 + 4 x2 + 3 x3 + 2 x4 + x5 + x6 6 and at most 3 variables are nonzero}. The following inequalities define facets of P:• 6 x1 + 4 x2 + 3 x3 + 2 x4 + x5 + x6 6• 4 x2 + 3 x3 + 2 x4 + x5 + x6 6• 4 x2 + 3 x3 + 2 x4 + x5 6• 4 x2 + 3 x3 + 2 x4 + x6 6• 4 x2 + 2 x4 + x5 + x6 6• 4 x2 + 2 x4 + x6 6• 4 x2 + 2 x4 + x5 6
To take advantage of previous work
… first, we scale and translate the variables,
i.e. P = conv {x [0, 1]n : jN aj xj b and xj
βj , j N, for at most k variables}, and
second, we consider the pieces of P
The pieces are defined as follows …
Proposition Let W N, XW = {x Rn : xj
βj j W and xj ≤ βj j N − W}, and PW = P ∩ XW . Then, PW = conv (S ∩ XW), where S = {x [0, 1]n : jN aj xj b and xj βj , j N, for at most k variables}. �
For each piece …
i.e. for a given W, we change the variables
as:
• yj ← (xj – βj) / (1 – βj), j W
• yj ← (βj – xj) / βj , j N − W
Example
P = conv {x [0, 1]2 : 6 x1 + 4 x2 7, and x1
=
½ or x2 = ½}.
½
½
1
1x1
x2
6 x1 + 4 x2 = 7
Example
P = conv {x [0, 1]2 : 6 x1 + 4 x2 7, and x1 =
½ or x2 = ½}.
½
½
1
1x1
x2
PN
P{1}
P{2}
P
Example
P = conv {x [0, 1]2 : 6 x1 + 4 x2 7, and x1 =
½ or x2 = ½}.
½
½
1
1x1
x2
3 y1 + 2 y2 2−3 y1 + 2 y2 2
3 y1 − 2 y2 2−3 y1 − 2 y2 2
at most 1 nonzero
at most 1 nonzeroat most 1 nonzero
at most 1 nonzero
When aj 0 and b > 0 …
Proposition The inequality jN xj k is facet-defining iff an−k + …+ an−1 b and a1 + an−k+2 + …+ an b. �
Proposition When an−k + …+ an−1 b and a1 + an−k+2 + …+ an > b, the inequality a1x1 +2≤j≤n−k−1 max {aj , Δ} xj +Δ n−k≤j≤n xj ≤ k Δ defines a facet of P, where Δ = (b − n−k−2≤i≤n
ai). �
It then follows that …
Proposition The inequality:
jW (xj – βj)/(1 – βj)–jN−W (xj – βj)/βj k is valid W N, and it is facet-defining “under certain conditions”. �
In the same way …
Proposition The inequality:
a1(x1 – β1)/(1 – β1) +2≤j≤n−k−1, jW max {aj ,
Δ}(xj – βj)/(1 – βj)+2≤j≤n−k−1, jN−W max {aj,
Δ} (xj – βj)/βj + Δ n−k≤j≤n , jW (xj – βj)/(1 – βj) + Δ n−k≤j≤n, jN−W (xj – βj)/βj ≤k Δ defines a facet of P “under certain conditions”. �
Example
P = conv {x [0, 1]2 : 6 x1 + 4 x2 7, and x1 =
½ or x2 = ½}.
½
1
1x1
x2
x1 + x2 3/2
x1 − x2 1/2x1 + x2 ≥ 1/2
(y1 + y2 1 and 3 y1 + 2 y2 2)
(y1 + y2 1 and 3 y1 − 2 y2 2)
−x1 + x2 1/2
(y1 + y2 1 and −3 y1 − 2 y2 2)
(y1 + y2 1 and −3 y1 + 2 y2 2)
Critical sets and covers
• By fixing, at 0 or 1, variables with positive or negative coefficients, we can obtain implied critical sets or cover inequalities that define facets in the projected polytope.
• Then, by lifting the fixed variables, we obtain strong inequalities valid for P
Example
Let P = conv {x [0,1]5: 6x1 + 4x2 − 3x3 − 2x4
+ x5 6 and at most 2 variables are positive}.
Fix x3 = 1 and x4 = 0. The inequality:
6x1 + 4x2 + 3x5 9
defines a facet of P ∩ {x [0,1]5: x3 = 1 and
x4 = 0}.
Simple bound inequalities
Let P = conv {x [0,1]4: 6x1 − 4x2 + 3x3 − x4
3 and at most 2 variables are positive}. Fix x3 = x4
= 0. Then, x1 1 defines a facet of P ∩ {x [0,1]4:
x3 = x4 = 0}. Lifting with respect to x4, we obtain x1 +
α x4 ≤ 1, which gives α = ⅓. Lifting now with
respect to x3, we obtain 3x1 + α x3 + x4 ≤ 3, which
gives α = 2, and so 3x1 + 2 x3 + x4 ≤ 3.
Additional results
• Two families of lifted cover inequalities
• Two families of inequalities derived from simple bounds
• Necessary and sufficient condition for “pieces of a facet” to be a facet
Further Research
• Separation routines and computational testing
• Inequalities derived from intersection of knapsacks
• Special results for feature selection in data mining