v2010.10.26 - Convex Optimization
v2010.10.26 - Convex Optimization v2010.10.26 - Convex Optimization
350 CHAPTER 4. SEMIDEFINITE PROGRAMMINGTwo interpretations of the constraints from problem (523) are realizedin Figure 104. Assuming that a cardinality-k solution exists and matrix Adescribes a pointed polyhedral cone K = {Ax | x ≽ 0} , columns are removedfrom A if they do not belong to the smallest face of K containing vector b .Columns so removed correspond to 0-entries in variable vector x ; in otherwords, generators of that smallest face always hold a minimal cardinalitysolution.Benefit accrues when vector b does not belong to relative interior of K ;there would be no columns to remove were b∈rel int K since the smallest facebecomes cone K itself (Example 4.5.1.8.1). Were b an extreme direction, atthe other end of our spectrum, then the smallest face is an edge that is a raycontaining b ; this is a geometrical description of a cardinality-1 case whereall columns, save one, would be removed from A .When vector b resides in a face of K that is not K itself, benefit isrealized as a reduction in computational intensity because the consequentequivalent problem has smaller dimension. Number of columns removeddepends completely on particular geometry of a given problem; particularly,location of b within K .There are always exactly n linear feasibility problems to solve in orderto discern generators of the smallest face of K containing b ; the topic of2.13.4.3. So comparison of computational ( intensity for this conic approachnpits combinatorial complexity ∝ a binomial coefficient versus n lineark)feasibility problems plus numerical solution of the presolved problem.4.5.1.8.1 Example. Presolving for cardinality-2 solution to Ax = b .(confer Example 4.5.1.5.1) Again taking data from Example 4.2.3.1.1, form=3, n=6, k=2 (A∈ R m×n , desired cardinality of x is k)⎡⎤ ⎡ ⎤A =⎢⎣−1 1 8 1 1 0−3 2 812−9 4 81413191− 1 2 31− 1 4 9⎥⎦ , b =⎢⎣11214⎥⎦ (682)proper cone K = {Ax | x ≽ 0} is pointed as proven by method of2.12.2.2.A cardinality-2 solution is known to exist; sum of the last two columns ofmatrix A . Generators of the smallest face that contains vector b , foundby the method in Example 2.13.4.3.1, comprise the entire A matrix becauseb∈int K (2.13.4.2.4). So geometry of this particular problem does not
4.5. CONSTRAINING CARDINALITY 351permit number of generators to be reduced below n by discerning thesmallest face. 4.37There is wondrous bonus to presolving when a constraint matrix is sparse.After columns are removed by theory of convex cones (finding the smallestface), some remaining rows may become 0 T , identical to other rows, ornonnegative. When nonnegative rows appear in an equality constraint to 0,all nonnegative variables corresponding to nonnegative entries in those rowsmust vanish (A.7.1); meaning, more columns may be removed. Once rowsand columns have been removed from a constraint matrix, even more rowsand columns may be removed by repeating the presolver procedure.4.5.2 constraining cardinality of signed variableNow consider a feasibility problem equivalent to the classical problem fromlinear algebra Ax = b , but with an upper bound k on cardinality ‖x‖ 0 : forvector b∈R(A)find x ∈ R nsubject to Ax = b‖x‖ 0 ≤ k(775)where ‖x‖ 0 ≤ k means vector x has at most k nonzero entries; such a vectoris presumed existent in the feasible set. Convex iteration (4.5.1) utilizes anonnegative variable; so absolute value |x| is needed here. We propose thatnonconvex problem (775) can be equivalently written as a sequence of convexproblems that move the cardinality constraint to the objective:minimizex∈R n 〈|x| , y〉subject to Ax = b≡minimize 〈t , y + ε1〉x∈R n , t∈R nsubject to Ax = b−t ≼ x ≼ t(776)minimizey∈R n 〈t ⋆ , y + ε1〉subject to 0 ≼ y ≼ 1y T 1 = n − k(525)4.37 But a canonical set of conically independent generators of K comprise only the firsttwo and last two columns of A.
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350 CHAPTER 4. SEMIDEFINITE PROGRAMMINGTwo interpretations of the constraints from problem (523) are realizedin Figure 104. Assuming that a cardinality-k solution exists and matrix Adescribes a pointed polyhedral cone K = {Ax | x ≽ 0} , columns are removedfrom A if they do not belong to the smallest face of K containing vector b .Columns so removed correspond to 0-entries in variable vector x ; in otherwords, generators of that smallest face always hold a minimal cardinalitysolution.Benefit accrues when vector b does not belong to relative interior of K ;there would be no columns to remove were b∈rel int K since the smallest facebecomes cone K itself (Example 4.5.1.8.1). Were b an extreme direction, atthe other end of our spectrum, then the smallest face is an edge that is a raycontaining b ; this is a geometrical description of a cardinality-1 case whereall columns, save one, would be removed from A .When vector b resides in a face of K that is not K itself, benefit isrealized as a reduction in computational intensity because the consequentequivalent problem has smaller dimension. Number of columns removeddepends completely on particular geometry of a given problem; particularly,location of b within K .There are always exactly n linear feasibility problems to solve in orderto discern generators of the smallest face of K containing b ; the topic of2.13.4.3. So comparison of computational ( intensity for this conic approachnpits combinatorial complexity ∝ a binomial coefficient versus n lineark)feasibility problems plus numerical solution of the presolved problem.4.5.1.8.1 Example. Presolving for cardinality-2 solution to Ax = b .(confer Example 4.5.1.5.1) Again taking data from Example 4.2.3.1.1, form=3, n=6, k=2 (A∈ R m×n , desired cardinality of x is k)⎡⎤ ⎡ ⎤A =⎢⎣−1 1 8 1 1 0−3 2 812−9 4 81413191− 1 2 31− 1 4 9⎥⎦ , b =⎢⎣11214⎥⎦ (682)proper cone K = {Ax | x ≽ 0} is pointed as proven by method of2.12.2.2.A cardinality-2 solution is known to exist; sum of the last two columns ofmatrix A . Generators of the smallest face that contains vector b , foundby the method in Example 2.13.4.3.1, comprise the entire A matrix becauseb∈int K (2.13.4.2.4). So geometry of this particular problem does not