Abstract This paper describes an algorithm for cardinality-constrained quadratic op-timization problems, which are convex quadratic programming problems with a limiton the number of non-zeros in the optimal solution. In particular, we consider prob-lems of subset selection in regression and portfolio selection in asset management andpropose branch-and-bound based algorithms that take advantage of the special struc-ture of these problems. We compare our tailored methods against CPLEX’s quadraticmixed-integer solver and conclude that the proposed algorithms have practical ad-vantages for the special class of problems we consider.
We present a method for solving cardinality-constrained quadratic optimization prob-lems (CCQO), i.e., quadratic optimization problems that limit the number of non-zero
The research of D. Bertsimas was partially supported by the Singapore-MIT alliance.The research of R. Shioda was partially supported by the Singapore-MIT alliance, the DiscoveryGrant from NSERC and a research grant from the Faculty of Mathematics, University of Waterloo.
D. BertsimasSloan School of Management and Operations Research Center, Massachusetts Instituteof Technology, E53-363, Cambridge, MA 02139, USAe-mail: [email protected]
R. Shioda (�)Department of Combinatorics and Optimization, Faculty of Mathematics, University of Waterloo,Waterloo, ON N2L 3G1, Canadae-mail: [email protected]
D. Bertsimas, R. Shioda
variables in the solution, using a tailored branch-and-bound implementation with piv-oting algorithms. Specifically, we consider the following problem:
minimize 12x�Qx + c�x,
subject to Ax ≤ b,
| supp(x)| ≤ K,
xi ≥ αi, i ∈ supp(x),
0 ≤ xi ≤ ui, i = 1, . . . , d,
(1)
where Q ∈ Rd×d is symmetric positive semi-definite, c ∈ R
d , A ∈ Rm×d , b ∈ R
m,αi > 0, ui is the nonnegative upper bound of xi , K is some positive integer, andsupp(x) = {i|xi �= 0}. The second set of constraints, referred to as the cardinalityconstraint, and the third set of constraints, referred to as the lower bound constraints,introduce discreteness to the problem, making this a quadratic mixed-integer opti-mization problem.
Compared to linear integer optimization, quadratic mixed-integer optimiza-tion problems have received relatively little attention in the literature. In the firststudy of problems of type (1), [4] proposes a tailored branch-and-bound algorithmand replaces the cardinality constraint | supp(x)| ≤ K with a surrogate constraint∑
i (xi/ui) ≤ K . Moreover, the x variables are branched on directly instead of in-troducing binary variables, i.e., the constraint xj ≤ 0 is added when branching downon xj and the constraint xj ≥ αi is added when branching up on xj . The underlyingquadratic solver used in [4] is a primal feasible algorithm that searches for feasibledescent directions, which includes Newton’s direction, steepest descent direction andFrank-Wolfe’s method. Warm-starting was done at each branch-and-bound node byusing a quadratic penalty function.
Motivated by this work, we extend the algorithm of [4] by using Lemke’s pivot-ing algorithm [7, 14] to solve the successive sub-problems in the branch-and-boundtree. Unlike [4], we do not explicitly add the variable bound constraints, xj ≤ 0 andxj ≥ αi , thus the size of the subproblems never increases. The other major distinc-tion to [4] is that we use a pivoting algorithm to solve the subproblems, which al-lows for efficient warm-starting. Section 2 elaborates on this general methodologyfor solving CCQO’s. In Sect. 3, we further tailor our method to solve two impor-tant problems in statistics and finance: subset selection in regression and portfolioselection in finance. We illustrate the results of our computational experiments inSect. 4.
2 General methodology
In a branch-and-bound setting, we solve the convex relaxation of Problem (1) viaLemke’s method, then choose a branching variable xs . When branching down, weupdate the subsequent subproblem by deleting the data associated to xs and whenbranching up, we modify Lemke’s method so that xs ≥ αs is enforced during pivoting.
Algorithm for cardinality-constrained quadratic optimization
The relaxation we solve at each node is:
minimize 12x�Qx + c�x,
subject to Ax ≤ b,
x ≥ 0,
xi ≥ αi, i ∈ U,
(2)
where the cardinality constraint is removed and U is the set of indices of variables thathave been branched up. The lower bound constraints xi ≥ αi for αi strictly positiveare enforced by implementing Lemke’s method with non-zero lower-bounds (anal-ogous to the simplex method with lower and upper-bounds). Section 2.1 describesthe use of Lemke’s method to solve Problem (2) in the context of branch-and-bound.Section 2.2 illustrates the procedure for updating the subproblem after the branchingvariable is deleted. Section 2.3 describes a heuristic based on our branch-and-boundprocedure for finding a good feasible solution.
2.1 Lemke’s method as underlying quadratic optimizer
We use Lemke’s pivoting method to optimize the convex relaxation of the sub-problem at each branch-and-bound node. This method was originally developed tosolve linear complementarity problems (of which quadratic programs are a specialcase) via pivoting akin to the simplex method. As with the dual simplex methodin linear optimization, the key advantage of Lemke’s method is its ease and effi-ciency of starting from an infeasible basic solution. This is critical in the branch-and-bound setting since the optimal solution of the parent node can be used as theinitial point to solve the problem of the current node. Thus, this approach has an ad-vantage over interior point methods which may need to solve from scratch at eachnode.
A linear complementarity problem (LCP) is the following: Given q ∈ Rn and M ∈
Rn×n, find z ∈ R
n and w ∈ Rn such that,
w = Mz + q, z ≥ 0, w ≥ 0, z�w = 0.
The above problem is referred to as LCP(q,M ). Clearly, the KKT necessary andsufficient optimality conditions of a convex quadratic programming problem is anLCP.
The Lemke’s method first checks whether q ≥ 0, in which case z = 0 and w = qis a solution. Otherwise, it augments LCP(q,M ) to
w = q + hz0 + Mz ≥ 0, z0 ≥ 0, z ≥ 0, z�w = 0, (3)
where h is some user-defined covering vector with h > 0. We need to start the al-gorithm with a complementary basis that does not necessarily satisfy the nonneg-ativity constraint. A simple default basis is to have all the z variables be nonba-sic and w be basic. We then set the auxiliary variable z0 to the smallest positivevalue such that w ≥ 0 when z = 0, i.e., z0 = maxi (−qi/hi), i = 1, . . . , d . Thus,ziwi = 0, i = 1, . . . , n and z0 > 0, and z0 is pivoted into the basis in place of wr ,
D. Bertsimas, R. Shioda
where r = argmaxi (−qi/hi). Such a point is called an almost complementary pointfor the augmented problem (3). The algorithm follows a path from one almost com-plementary basic solution to the next, until z0 is pivoted out to be a nonbasic variableor LCP(q,M ) is shown to be infeasible [13].
During the branch-and-bound procedure, we want to resolve a new subproblemstarting with the basic solution of the parent subproblem. Let M and q be themodified data for the current subproblem, and let B and N be the correspondingcolumns of the basic and nonbasic variables, respectively, of the parent node. We want
Lemke’s method to solve LCP(M B , q B ), where M B = −B−1
N and q B = B−1
q ,and have B as its initial complementary basis matrix. This basis is most likely notfeasible for LCP(M B , q B ), thus the problem is augmented by the auxiliary variablez0, z0 is increased until the initial basis is feasible for the augmented problem, andthen we execute sequence of pivots until z0 is pivoted out or the LCP is deemedinfeasible.
2.2 Branching down
When branching down on xs , we delete all the data associated to xs for thesubsequent subproblems and update the basis accordingly. We chose to deletethe variable instead of explicitly adding the constraint xs = 0 to prevent in-creasing the size of the subproblem as well as for numerical stability purposes.We will show that in most cases, the inverse of the new basis can be effi-ciently derived from the inverse of the old basis via elementary row opera-tions.
Let us assume that xs is a basic variable and suppose B and N are basic andnonbasic columns, respectively, of the previous solution. We delete the column androw of B corresponding to xs and the column and row of N corresponding to itsdual variable ws . Although we can get the new inverse of the basis simply by in-verting the modified basis, calculating the inverse can be a significant bottleneck.Instead, we calculate the new inverse from the previous inverse using elementary rowoperations.
Suppose, for notational purposes, that the column and row needed to be deleted inB are the first column and first row and B ∈ R
n×n, so that:
B =[
v B�row
Bcol B
]
, B−1 =[
u U�row
U col U
]
,
where B and U are n − 1 by n − 1 lower-right submatrices of B and B−1, respec-tively, Bcol, Brow, U col and U row are (n − 1)-dimensional column vectors, and v andu are scalars. We know that
B−1B =[
uv + U�rowBcol uB�
row + U�rowB
vU col + UBcol U colB�row + UB
]
= I n,
thus,
UB = I n−1 − UcolBrow�. (4)
Algorithm for cardinality-constrained quadratic optimization
Since U colB�row is a rank one matrix, we can execute linear number of elemen-
tary row operations to the matrix I n−1 − U colB�row to get I n−1. Let E be the ma-
trix representing those operations. Then EU is the inverse matrix of B if B isinvertible.
In the previous section, we stated that we use M B = −B−1
N as input toLemke’s. We avoid this matrix multiplication via similar elementary row oper-ations. Suppose MB = −B−1N at termination of Lemke’s at the parent node.Again, let us assume that the column corresponding to ws in N is the first one.Then,
MB = −B−1N = −[
u U�row
U col U
][p N�
row
Ncol N
]
=[
w M�row
Mcol M
]
,
where N and M are n − 1 by n − 1 lower-right submatrices of N and MB , respec-tively, and N col, N row, Mcol and M row are (n − 1)-dimensional column vectors, andp and w are scalars. Again, we know that
−UN = M + U colN�row.
Since EU = B−1
, the new M B matrix will be
M B = E(M + U colN�row). (5)
There are several assumptions that need to be checked before executing theabove procedures. Most critically, if B is singular, then E may be undefined. Insuch a case, we start Lemke’s method from scratch with the initial basis B =I n−1. Clearly, this is not the only solution to this problem, but the scenario oc-curred rarely enough in practice so that this method was adequate for our pur-poses. Also, we assumed that we deleted the first row and nth column from B ,B−1, N and MB . The general case can be easily modified to this special case.Finally, if xs is a nonbasic variable in the previous solution, we can apply thefollowing methodology for its complementary variable ws which must be a basicvariable.
To update qB , we delete the sth element of c, giving us c. Suppose qB = B−1q atthe termination of Lemke’s method, where q = [ c
b
]. Again, assuming that s = 1, we
have:
qB =[
qs
qB
]
= B−1q =[
u U�row
U col U
][qs
q
]
,
where qB and q is the (n−1) lower subvector of qB and q , respectively, and qs = cs .Similar to M B , we get:
q B = E(qB − qsUcol). (6)
LU decomposition of the basis
From (5) and (6), we only need to know one column of B−1 to update M and q . Thus,instead of explicitly maintaining B−1, we calculate the LU decomposition of the
D. Bertsimas, R. Shioda
basis B at the termination of Lemke’s method and use it only to derive the requiredcolumn of B−1.
We use Crout’s algorithm [22] to construct the LU decomposition of B and derivethe sth column of B−1 using back-substitution. If xs is the ith basic variable, thenwe get U col by deleting the ith element of the column. Given B , N and U col, we canupdate M and q according to (5) and (6), respectively.
2.3 A heuristic
To find a good feasible solution at the root node, we run a heuristic which com-bines the heuristics proposed by [4] and [12]. Let x∗ be the solution of the con-tinuous relaxation at the root node. We first run what [12] refers to as the “reop-timization heuristic” which “reflects common practice”, where the continuous re-laxation is solved again using only the variables with the K largest absolute valuesof x∗. The lower-bound constraint xi ≥ αi is also imposed for these variables. Ifthis problem is feasible, the solution is a feasible solution to the original CCQOand let UB0 be its corresponding objective value. To improve on UB0, we thenrun the heuristic proposed by [4]. Let G = {i | |x∗
i | is one of the K + W largestabsolute values of x∗}, where W is a user-defined small positive integer such that|G| = K + W � d (we have set W = 0.1d in our computational experiments). Wethen solve the CCQO problem using only the variables in G, setting UB0 as the ini-tial upper-bound to the optimal value. We also put a limit on the number of nodesto examine. Thus, we are implementing our branch-and-bound procedure just on thevariables in G.
3 Applications of CCQO
We focus on applying the methodology described in Sect. 2 to the K-subset selectionproblem in regression and optimal portfolio selection in Sects. 3.1 and 3.2, respec-tively.
3.1 Subset selection in regression
In traditional multivariate regression, we are given m data points (xi , yi), x i ∈ Rd ,
yi ∈ R, and we want to find β ∈ Rd such that
∑i (yi − x�
i β)2 is minimized. Thishas a closed-form solution, β = (X�X)−1X�Y , where X ∈ R
m×d with x�i as its
ith row, and Y ∈ Rm with yi as its ith element. Primarily for robustness purposes,
i.e., to limit the variance of the predicted Y , it is desirable to use a small subset ofthe variables (see [1, 18, 23]). For example, suppose we want to choose K variables(K < d) that minimizes the total sum of squared errors. We formulate this prob-lem as:
minimize (Y − Xβ)�(Y − Xβ),
subject to | supp(β)| ≤ K,(7)
which is clearly a CCQO.
Algorithm for cardinality-constrained quadratic optimization
Authors of [2, 9, 10] use pivoting and enumeration to search for subsets withthe best regression “fit” (e.g., minimum total sum of squared errors) for sub-sets of all sizes. Authors of [19] solve linear mixed-integer optimization problemsthat find a subset of K variables (K < d) that has the minimum total absoluteerror.
We solve (7) by tailoring our approach to this unconstrained version of a CCQO.When the cardinality constraint is relaxed, the optimal objective value is Y�Y −Y�X�(X�X)−1X�Y , thus we do not need to run the Lemke’s method. The maincomputational work is in the branch down procedure. Clearly, we can extend ourmethod to regression problems with linear constraints with respect to the β’s byapplying our general methodology. However to highlight the merits of our tailoredapproach versus a general purpose software such as CPLEX, we focus on the uncon-strained regression in this paper.
When branching down on xs , we delete the sth row and column of X�X, and theinverse (X�X)−1 is updated as illustrated in Sect. 2.2. To further alleviate computa-tion, we set v = X�Y ∈ R
d at the root node. Thus, deleting xs corresponds to deletingthe sth element of v. We do not need to multiply X� and Y in subsequent nodes—wesimply need to delete corresponding elements from v. The optimal objective value ofa given node is then:
Y�Y − v�(X�X)−1v,
where X�X and v are the updated X�X and v, respectively. Thus, calculating theobjective value requires only matrix-vector multiplications. There is no need to up-date the subproblem when branching up, since the optimal solution of the parentnode is optimal for the next node. Section 4.1 illustrates computational results of ourapproach.
3.2 Portfolio selection
Let us consider the traditional mean-variance portfolio optimization problem, whichcan be modeled as a convex quadratic optimization problem. Large asset manage-ment companies manage assets against a benchmark that has d securities. So thatthey are not seen as indexers, it is desirable that they only use K � d securitiesin the portfolio. In addition, several portfolios are marketed as focused funds thatare only allowed by their prospectus to own a small collection of securities. Fi-nally, asset management companies that manage separate accounts for their clientsthat only have say $100,000 can only realistically own a small number of securities,since otherwise, the transaction costs would significantly affect performance. Suchlimited diversification constraints, along with fixed transaction costs and minimumtransaction levels, present discrete constraints and variables to the quadratic problem(see [3]).
Given the difficulty of solving quadratic integer optimization problems, sucha portfolio problem has commonly been approached in one of two ways. Thefirst approach is approximating the problem to a simpler form. For example,[24, 25] approximate the quadratic objective function by a linear and piece-wiselinear function, and [11] further assumes equal weights across assets to formulate
D. Bertsimas, R. Shioda
the problem as a pure 0-1 problem. In [21], portfolio problems with fixed trans-action costs are solved in polynomial time when the covariance matrix has equaloff-diagonal elements. The second approach uses heuristics to find strong feasiblesolutions. For example, [17] proposes a linear mixed-integer optimization basedheuristic, [5] introduces a dynamic programming based heuristic, and [6] pro-poses genetic algorithm, tabu search and simulated annealing approaches for theproblem.
There have been also significant efforts in finding exact algorithms to solve dis-crete portfolio optimization problems. For example, [20] extends the work of [4] tolimited diversification portfolios, and [12] solves portfolio problems with minimumtransaction levels, limited diversification and round lot constraints (which requiresinvesting in discrete units) in a branch-and-bound context. In [15], the authors solvea portfolio optimization problem that maximizes net returns where the transactioncosts are modeled by a concave function. They successively estimate the concavefunction by a piecewise linear function and solve the resulting LP. They have shownthat their solution converges to the optimal solution of the original problem. In [16],the authors present a divide-and-conquer algorithm that partitions the feasible set ofthe solutions to find the exact solution to a problem with fixed transaction costs andround lots.
In this paper, we focus on the traditional mean-variance portfolio optimizationmodel with cardinality constraints. The key difference between our approach andthose described above is our use of Lemke’s pivoting algorithm to solve the underly-ing quadratic program in our branch-and-bound implementation. Let us further sup-pose that the d stocks can be categorized into S industry sectors. Investors may wishto limit the total investment change within each of S industry sectors. Let the currentportfolio weights be x0 ∈ R
d . The traditional mean-variance model determines thenew weights for the d stocks, x ∈ R
d , that maximizes the total expected return minusa penalty times total variance. In practice, there are other direct and indirect transac-tion costs, such as price impact costs and ticket costs. Impact costs reflect the stockprice impact resulting from purchase or sale orders and the magnitude of this costdepends on the particular stock and the trade sizes. For example, large purchase or-ders will increase the price and large sales orders will decrease the price of the stock.Assuming symmetric impact for purchases and sales, this effect is often modeled bythe following quadratic function
d∑
i=1
ci(xi − x0i )2,
where ci > 0 is the impact coefficient for Stock i. The second form of transactioncosts is ticket cost, which is a fixed cost associated to trading a positive volume of astock. This cost can easily be incorporated into our CCQO framework using binaryvariables, but we will not include it in the present work because ticket costs are of-ten second order compared to impact costs. This portfolio selection problem can be
Algorithm for cardinality-constrained quadratic optimization
represented by the following formulation:
minimize −r�x + 12 (x − xB)��(x − xB) +
d∑
i=1
ci(xi − x0i )2,
subject to
∣∣∣∣
∑
i∈Sl
(xi − xBi )
∣∣∣∣ ≤ εl, l = 1, . . . , S,
d∑
i=1
xi = 1,
| supp(x)| ≤ K,
xi ≥ αi, i ∈ supp(x),
xi ≥ 0, i = 1, . . . , d,
(8)
where � is the covariance matrix of the rates of return, xBi is the benchmark weight
for stock i, r is a d-dimensional vector of the expected rates of return, αi is theminimum transaction level of stock i, ci is the price impact coefficient for Stock i,and Sl is the set of indices of stocks in Sector l. The first constraint limits the totalchange in the portfolio weights in Sector l to be less than some εl . The second setof constraints ensures that the weights sum up to 1, and the third constraint limitsinvesting up to K stocks. The fourth set of constraints implies that if we invest instock i, then xi must be at least αi . Clearly, Problem (8) is in the form of Problem (1)and can be solved using our methodology.
We rewrite Problem (8) as
minimize −r�x + 12x��x + C0,
subject to∑
i∈Sl
xi ≤ εl +∑
i∈Sl
xBi , l = 1, . . . , S,
−∑
i∈Sl
xi ≤ εl −∑
i∈Sl
xBi , l = 1, . . . , S,
d∑
i=1
xi = 1,
| supp(x)| ≤ K,
xi ≥ αi, i ∈ supp(x),
xi ≥ 0, i = 1, . . . , d,
(9)
where r = r +�xB + 2Cx0, � = � + 2C where C is the diagonal matrix with ci asthe ith diagonal element, and C0 is a constant equal to (xB)��xB + ∑d
i=1 ci(x0i )2.
We solve the relaxation of Problem (9) using Lemke’s method described inSect. 2.1, and branch down on a variable as in Sect. 2.2. Section 4.2 illustrates thecomputational results of this method.
D. Bertsimas, R. Shioda
4 Computational results
We describe computational experiments on subset selection and portfolio selectionproblems in Sects. 4.1 and 4.2, respectively. For each problem, we compare ourtailored approaches to CPLEX’s quadratic mixed integer programming solver [8].Clearly, CPLEX’s implementation of the pivoting methods and branch-and-bound isfar superior to ours, however, our motive is to measure the advantages of a tailoredimplementation over a general mixed-integer solver for these particular CCQO prob-lems.
4.1 Results for subset selection
We compared our branch-and-bound implementation for solving subset selectionwith forward regression and CPLEX’s quadratic mixed-integer programming solver.Forward regression is a greedy heuristic that, given the variables already chosen,chooses another variable that reduces the residual error the most, i.e., the firstvariable chosen, β(1), corresponds to β(1) = arg minj=1,...,d
∑i (yi − xi,j βj )
2. Thenext variable minimizes
∑i (yi − xi,j βj )
2 for all j ∈ {1, . . . , d} \ (1), where yi =yi − xi,(1)β(1). This step is repeated until K variables are chosen [18].
We used CPLEX’s quadratic mixed-integer optimizer to solve Problem (7) byintroducing binary inclusion variables zi , replacing the cardinality constraint by∑
i zi ≤ K and adding constraints βi ≥ −Mzi and βi ≤ Mzi , where M is some largepositive number. We found that setting M = 100 was sufficiently large to solve ourgenerated problems effectively. By comparing our method with CPLEX, we hope tosee the computational advantages, if any, of not using binary variables zi and branch-ing directly on βi ’s. One obvious benefit is the elimination of the need for the socalled “big-M” constraints.
Our branch-and-bound and CPLEX’s branch-and-bound search procedure useddepth-first-search, and branches on the variable with maximum absolute value first. Inour algorithm, we ran the heuristic presented in Sect. 2.3 after solving the continuousrelaxation of the root node and not in any subsequent nodes.
For each d (the number of variables), we randomly generated five instances of X
and β , and set Y = Xβ + ε , where εi ∼ N(0,1) for each i. For each problem size,we present the average performance of the methods over all five instances in Tables 1and 2. The performances of the individual instances are presented in Tables 6 and 7 inthe Appendix. In all of the tables, the columns “Forward”, “BnB”, and “CplexMIQP”correspond to the results of the forward regression, our method, and CPLEX, respec-tively. We did not record the running time for forward regression since this simpleheuristic was able to solve almost all the instances in a fraction of a second. The col-umn labeled “time” is the total CPU seconds required to solve the problem up to aspecified time limit (discussed below). This number includes the running time of theroot node heuristic as well. The column labeled “nodes” is the total number of nodesin the branch-and-bound tree at termination, “best node” is the node correspondingto the best feasible solution and “RSS” is the best total sum of squared errors foundfor subsets of size K . All the numbers, except for the CPU time, were rounded to thenearest integer.
Algorithm for cardinality-constrained quadratic optimization
Table 1 Results for Subset Selection with 60 CPU seconds. The column “time” is in CPU seconds and“RSS” is the residual sum of squares
Table 1 shows the results when the time limit for both our method and CPLEXwas set 60 CPU seconds. These results illustrate whether these exact methods canfind a “good” feasible solution relatively quickly. It is apparent from this table thatthe exact approaches significantly improve upon the forward regression in terms ofresidual sum of squares, even when they do not solve to provable optimality. Both“BnB” and CPLEX solved the problems with (d,K) = (20,10), (20,5), (50,40) toprovable optimality in several seconds. However, in all cases where CPLEX does notfind the optimal solution within 60 seconds, the RSS of “BnB” is consistently lowerthan that of CPLEX. This is most evident in cases where K is large relative to d , i.e.,for (d,K) = (100,80) and (d,K) = (500,400). CPLEX performs especially poorlyin these cases, having RSS values worse than that of the forward heuristic. In contrast,“BnB” appears to do especially well, having significantly lower RSS values thanthe other two methods. For three out of the five instances with (d,K) = (100,80),
D. Bertsimas, R. Shioda
our method found the provably optimal solution in under one second (see Table 6).From the “best node” column, it is clear that our heuristic, which applies our branch-and-bound procedure on a smaller subproblem of the original CCQO, yields goodsolutions. The CPLEX routine also runs a general heuristic at the root node, but itdoes not appear to be as effective for these problems.
These results show that in a minute or less, our method is able to provide solutionsthat are often substantially better than that of the forward heuristic. Thus, if speed isimportant, using our method with a short time limit may be a viable alternative to theforward heuristic.
Table 2 illustrates the results when our method and CPLEX are run for 3600 CPUseconds. For (d,K) = (50,20), both “BnB” and CPLEX solved to provable optimal-ity within minutes, where “BnB” had faster running times in three out of the five in-stances. Our method also solves all five instances of (d,K) = (100,80) in an averageof 8 minutes, however, it is surprising to see that CPLEX could not solve three out ofthe five instances within one hour (two of them being instances that “BnB” solved inunder one minute—see Table 7). CPLEX solved the remaining two instances in about45 minutes each. As in Table 1, CPLEX performs relatively poorly when K is large,namely for (d,K) = (100,80) and (500,400). It is not clear why these instances areespecially difficult for CPLEX to find a good feasible solution. Again, in every sin-gle instance where CPLEX did not find the optimal solution, the best solution foundby “BnB” was superior to that of CPLEX. We also see that our heuristic solution isstill very strong. With the longer running time, our method is able to improve on theheuristic solution, however, it is often very close to the best RSS value found in onehour.
We selected some problem instances and ran CPLEX for up to 12 hours in hopesof finding a provably optimal solution. However, CPLEX could not find a better so-lution than those found by “BnB” in one hour. For example, in one of the instanceof (d,K) = (100,20) (this is problem instance (d,K,v) = (100,20,1) in Table 7of the Appendix), CPLEX could not find the optimal solution in 12 hours. Its bestRSS value was 758,261 (which was found by its root node heuristic) whereas theRSS value found by “BnB” in one hour was 690,532. In one of the instance of(d,K) = (500,400) (this is problem instance (d,K,v) = (500,400,1) in Table 7of the Appendix), CPLEX’s best RSS value was 56,390,541, whereas the RSS valuefound by “BnB” in one hour was 38,781. Thus, it appears that this general solver isnot well suited to solved this particular type of CCQO.
The difference in the two methods is most likely due to the difficulty of solving theexplicit mixed-integer quadratic program formulation of the subset selection problemwith the binary variables and big-M constraints. These big-M constraints can lead toweak LP relaxations, making it hard to fathom nodes. In addition, our branch-and-bound based heuristic appears to be very effective in finding strong feasible solutions.This combination allows us to fathom many more nodes than CPLEX.
Note that the per node computation time for “BnB” decreases as K decreases.This is highlighted in d = 100 and d = 500, where the average number of nodesexplored increases significantly as K decreases. This is because we delete variablesthat are branched down, making the subproblem smaller and smaller as we go downthe branch-and-bound tree until at most d − K variables are deleted. Thus, many of
Algorithm for cardinality-constrained quadratic optimization
the subproblems solved when K = 20 is much smaller than when K = 400, resultingin the difference in average per node computation time.
The main bottleneck of our method is the number of nodes needed to prove op-timality. Even when the heuristic finds the optimal solution at the root, the branch-and-bound tree can grow to a million nodes to prove optimality, even for moderatesized problems. The main factors preventing significant pruning of the tree are thefree variables and the lack of constraints. A subproblem solution almost always hasall non-zero variables. However, provable optimality may not be of critical impor-tance to data mining practitioners who may be interested in finding a “good” solution“quickly”. From these empirical studies, we have seen that our branch-and-boundprocedure finds near-optimal solutions in one minute. Given the noise in the data,perhaps these solutions are sufficient in practice.
4.2 Results for portfolio selection
We tested our approaches described in Sect. 3.2 against two alternative methods.One method uses CPLEX’s quadratic barrier method to solve the relaxation prob-lem of (9), and the second uses CPLEX’s quadratic mixed-integer solver to solvethe explicit mixed-integer formulation. All branch-and-bound procedures, includingCPLEX, were set to depth-first-search, branch up first and branch on variable withmaximum absolute value. Depth-first-search was used primarily due to its ease ofimplementation and limited memory usage.
For each d (total number of assets), S (number of sectors), and K (the upper boundon diversification), we generated five random instance of Problem (8) and averagedthe results. We set x0 = 0 for all of our instances. For �, we generated a matrix A
from a Gaussian distribution, then used dd−1AT A as our covariance matrix. The val-
ues of r and ci were taken from a Gaussian and uniform distribution, respectively. Weacknowledge that using randomly generated data may give unrealistically optimisticrunning times. However, since our interest is to compare the computational perfor-mance of different solution methods, we hope that this relative difference extends toreal stock data as well. Tables 3 and 4 illustrate the results. In these tables, “UB” isthe best feasible solution found, “best node” is the node where “UB” was found and“nodes” is the total number of nodes explored. Entries labeled “-” indicate that themethod failed to find a feasible solution within 120 CPU seconds.
“LemkeBnB” refers to our method described in Sect. 2. “BarrierBnB” is the sameas “LemkeBnB”, except we use CPLEX’s barrier method to solve the continuousquadratic optimization problem. Finally, “CplexMIQP” is the result of using CPLEXquadratic mixed-integer solver. All three methods were run for a total of 120 CPUseconds. The column labeled “nodes” is the average of the total number of nodeseach method explored, “best node” is the average of the node where the best feasiblesolutions were found, and “UB” is the best feasible objective value found within thetime limit.
Table 3 runs all three methods without running any heuristic methods to finda good feasible solution, whereas Table 4 runs a heuristic once at the root. For“LemkeBnB” and “BarrierBnB”, we ran the heuristic for at most 30 CPU secondsafter the root is solved. We were not able to put a time limit on the heuristic for
D. Bertsimas, R. Shioda
Table 3 Results for portfolio selection, without Heuristic, solved until 120 CPU seconds
d K S LemkeBnB BarrierBnB CplexMIQP
Nodes Best node UB Nodes Best node UB Nodes Best node UB
CplexMIQP, which ran until CPLEX deemed it had an acceptable upper-bound.When the size of K was relatively large (K > 0.1d), the CPLEX heuristic ran forat most 10 CPU seconds, whereas it can last over 100 CPU seconds when d is large(d > 200) and K is small compared to d (e.g., K ≤ 0.1d).
Both pivoting-based methods, “LemkeBnB” and “CplexMIQP”, are significantlyfaster than “BarrierBnB” for every instance. Although the relative difference in thetotal number of nodes explored did decrease as the problem size increased, the ad-vantage of interior point methods in large dimensions over pivoting methods did notcompensate the latter’s advantage in warm starting. For example, for problems thatwould take an average of 400 pivots to solve from scratch, any intermediary nodewould require only about 5 pivots to resolve the subproblem of that node. Also, thepivoting methods always give a basic feasible solution to the KKT equations of thequadratic programming problem. Thus, it is guaranteed to give the solution to therelaxation with the minimum support, unlike the interior point method. Since manyof the generated instances and most real world problems do not guarantee positivedefiniteness (only positive semi-definiteness) in the quadratic matrix, this differencecan be another significant advantage.
It is clear that the node to node computation time of “CplexMIQP” is faster than“LemkeBnB” for most instances. However, the relative difference significantly de-creases as K becomes small relative to d . For example, when K ≈ 0.5d , the differ-
Algorithm for cardinality-constrained quadratic optimization
Table 5 Results for portfolio selection, with the root Heuristic, solved for 3600 CPU seconds
d K S LemkeBnB CplexMIQP
Nodes Best node UB Nodes Best node UB
500 200 10 6,100 0 83.93 49,720 0 138.83
500 100 10 5,960 0 80.40 55,183 0 140.29
500 50 4 13,828 0 80.01 15,078 7,995 89.34
ence in number of nodes explored is about a factor of 10, whereas when K ≈ 0.1d , itreduces to a factor of 2 to 3. For the case when d = 200 and K = 20, the total compu-tation time of “LemkeBnB” and “CplexMIQP” are about the same. We reach similarconclusions when we increase the running time. Table 5 shows results for runningour method and CPLEX for 3600 CPU seconds.
Running the heuristic, even for just 30 CPU seconds, brought significant improve-ment to our models in terms of finding good feasible solutions. Using |G| smallenough so that Lemke’s method can run sufficiently fast allowed us to find a goodfeasible solution quickly.
5 Conclusion
From this computational study, we learn that:
1. Our tailored approaches for solving cardinality constrained quadratic optimiza-tion problems in regression and portfolio optimization show some computationaladvantages over a general mixed-integer solver.
2. For subset selection in regression, our method was able to find the subset of vari-ables with significantly better fit than the forward regression heuristic even with a60 second time limit. The results also show that our approach has significant com-putational advantages to CPLEX which needs to solve an explicit mixed-integerquadratic formulation with big-M constraints.
3. For the portfolio selection problem, the combination of our branch-and-boundimplementation and Lemke’s method has significantly faster running times com-pared to using the barrier method to solve the continuous quadratic optimizationproblem. The key bottleneck for efficient quadratic mixed-integer optimizationhas been the inability of interior point methods to start at infeasible points. Al-though they are undoubtedly more effective in solving high dimensional quadraticoptimization problems than pivoting methods started from scratch, the pivotingmethods can re-solve each subproblem more efficiently at each node of the branch-and-bound tree.
4. CPLEX’s quadratic mixed-integer solver has a more sophisticated pivoting andbranch-and-bound implementation, yet our tailored approach compensates forour lack of software engineering prowess. Our root heuristic finds good upperbounds quickly and our variable deletion and Lemke’s method with non-zerolower bounds updates each subproblem without increasing the size of the prob-lem. With further improvements in implementation (e.g., regarding data struc-
D. Bertsimas, R. Shioda
tures, decompositions, and memory handling), we believe our methodology willhave comparable node-to-node running times.
There are several potential improvements to our model. We use depth-first-search inall of our branch-and-bound procedures due to the ease of implementation. Althoughwe can find good upper-bounds faster with this approach, we often get stuck in asubtree. Also, with large number of nodes, we cannot utilize best lower-bounds ef-fectively, since the root relaxation would be the lower-bound for a large majority ofthe nodes we explore. There is also merit in investigating other principal pivotingtechniques other than Lemke’s. The main drawback of Lemke’s method is the lack ofchoice in the entering variable. Alternative pivoting methods, though more difficultto initialize, have the flexibility of choosing amongst several entering variables, akinto the simplex method. It also does not augment the LCP, thus not introducing anauxiliary variable and column. These properties may allow us to converge faster tothe solution.
Our goal in this work was to investigate the computational merit of tailoredbranch-and-bound implementations that does not require introducing binary variablesfor solving subset selection in regression and portfolio selection problem in assetmanagement. To find good (but not necessarily provably optimal) solutions quickly,these approaches appear to have advantages over generalized solvers. We hope tofurther improve our implementation and explore the practicality of our algorithm toother examples of CCQOs.
Algorithm for cardinality-constrained quadratic optimization
App
endi
xA
:A
ddit
iona
ltab
les
Tabl
e6
Res
ults
for
Subs
etSe
lect
ion
with
60C
PUse
cond
s.d
isth
enu
mbe
rof
vari
able
s,K
isth
esi
zeof
the
sele
cted
subs
et,a
ndR
SSis
the
resi
dual
sum
ofsq
uare
s.Fi
vedi
ffer
enti
nsta
nces
for
each
(d,K
)pa
irw
ere
solv
edan
dv
deno
tes
the
inst
ance
num
ber
dK
vFo
rwar
dB
nBC
plex
MIQ
P
RSS
CPU
sec.
#no
des
Bes
tnod
eR
SSC
PUse
c.#
node
sB
estn
ode
RSS
2010
15,
730
0.08
757
481
4,88
20.
1077
636
64,
881
2010
212
,391
0.01
690
5,22
30.
0311
891
5,22
3
2010
35,
572
0.00
270
2,73
30.
0250
492,
733
2010
46,
540
0.04
283
994,
018
0.05
268
196
4,01
8
2010
57,
356
0.00
330
4,67
30.
0232
04,
673
205
114
,406
0.00
570
11,2
750.
0210
110
011
,275
205
226
,221
0.02
227
025
,162
0.07
408
364
25,1
62
205
326
,333
0.02
177
018
,658
0.04
219
191
18,6
58
205
411
,440
0.01
115
011
,440
0.04
196
174
11,4
40
205
533
,392
0.03
295
1030
,181
0.06
406
404
30,1
81
5040
166
,070
0.13
850
489
14.8
631
,631
31,6
3048
9
5040
227
,970
8.52
5,12
50
6,43
326
.06
71,5
2170
,266
6,43
3
5040
313
,662
0.12
830
465
21.6
445
,194
45,1
9346
5
5040
411
,864
0.14
930
2,17
55.
4112
,497
12,4
972,
175
5040
567
,712
0.53
357
03,
190
7.34
18,1
8017
,905
3,19
0
D. Bertsimas, R. Shioda
Tabl
e6
(Con
tinu
ed)
dK
vFo
rwar
dB
nBC
plex
MIQ
P
RSS
CPU
sec.
#no
des
Bes
tnod
eR
SSC
PUse
c.#
node
sB
estn
ode
RSS
5020
194
,662
60.0
088
,631
047
,481
60.0
016
6,66
90
55,0
29
5020
217
3,33
060
.00
88,1
470
99,1
5960
.00
168,
326
010
1,61
4
5020
310
1,75
760
.00
90,0
1737
,975
66,0
5160
.00
167,
051
081
,622
5020
470
,347
60.0
087
,941
052
,589
60.0
016
7,05
10
53,4
37
5020
512
7,48
060
.00
88,9
070
61,2
2360
.00
166,
943
069
,659
100
801
339,
592
60.0
04,
035
016
,108
60.0
053
,260
52,6
4872
2,84
1
100
802
410,
155
0.82
161
095
460
.00
53,9
8551
,888
626,
166
100
803
376,
343
60.0
07,
289
6,37
713
,899
60.0
054
,479
54,4
5066
4,73
4
100
804
302,
706
0.81
161
096
560
.00
53,4
6352
,927
761,
439
100
805
369,
136
0.96
163
01,
126
60.0
052
,856
52,8
0459
6,22
0
100
501
611,
976
60.0
09,
193
012
9,70
660
.00
54,5
170
758,
261
100
502
461,
085
60.0
08,
929
8,74
110
4,64
460
.00
54,7
380
630,
966
100
503
443,
940
60.0
09,
213
013
1,32
760
.00
55,0
020
678,
234
100
504
379,
711
60.0
09,
165
6311
1,73
860
.00
54,8
260
776,
809
100
505
538,
908
60.0
09,
013
2,88
510
3,91
260
.00
54,6
680
618,
520
100
201
918,
579
60.0
035
,619
25,3
8569
0,53
260
.00
52,7
730
758,
261
100
202
807,
784
60.0
033
,031
059
1,70
060
.00
53,2
730
630,
966
100
203
758,
718
60.0
031
,655
062
2,72
860
.00
54,0
000
678,
234
100
204
793,
692
60.0
033
,889
071
3,57
660
.00
53,6
940
776,
809
100
205
796,
714
60.0
037
,931
056
4,57
860
.00
53,9
940
618,
520
Algorithm for cardinality-constrained quadratic optimization
Tabl
e6
(Con
tinu
ed)
dK
vFo
rwar
dB
nBC
plex
MIQ
P
RSS
CPU
sec.
#no
des
Bes
tnod
eR
SSC
PUse
c.#
node
sB
estn
ode
RSS
500
400
111
,307
,900
60.0
03
039
,566
60.0
032
00
64,0
22,3
00
500
400
211
,811
,700
60.0
03
012
5,12
560
.00
591
590
120,
308,
000
500
400
315
,760
,100
60.0
03
076
,127
60.0
02
011
7,63
9,00
0
500
400
411
,463
,100
60.0
03
04,
645
60.0
036
70
58,8
63,4
00
500
400
59,
227,
680
60.0
03
075
,680
60.0
044
80
54,4
04,7
00
500
100
121
,815
,400
60.0
03
015
,267
,300
60.0
012
00
64,0
22,3
00
500
100
222
,578
,200
60.0
03
014
,331
,100
60.0
057
556
412
0,68
2,00
0
500
100
325
,562
,500
60.0
03
017
,802
,300
60.0
00
011
7,63
9,00
0
500
100
421
,751
,200
60.0
03
015
,591
,100
60.0
038
30
58,8
63,4
00
500
100
519
,525
,800
60.0
03
014
,681
,000
60.0
043
00
54,4
04,7
00
500
201
34,0
13,8
0060
.00
1,45
179
532
,687
,900
60.0
025
50
64,0
22,3
00
500
202
36,1
03,3
0060
.00
1,67
545
733
,350
,000
60.0
00
013
3,19
8,00
0
500
203
42,8
41,9
0060
.00
1,60
11,
017
41,5
50,0
0060
.00
120
117,
639,
000
500
204
36,6
22,5
0060
.00
1,23
381
135
,780
,100
60.0
038
40
58,8
63,4
00
500
205
34,1
51,1
0060
.00
1,48
31,
259
33,9
24,4
0060
.00
00
136,
429,
000
D. Bertsimas, R. Shioda
Tabl
e7
Res
ults
for
Subs
etSe
lect
ion
with
3600
CPU
seco
nds.
dis
the
num
ber
ofva
riab
les,
Kis
the
size
ofth
ese
lect
edsu
bset
,and
RSS
isth
ere
sidu
alsu
mof
squa
res.
Five
diff
eren
tins
tanc
esfo
rea
ch(d
,K
)pa
irw
ere
solv
ed.v
deno
tes
the
inst
ance
num
ber
dK
vFo
rwar
dB
nBC
plex
MIQ
P
RSS
CPU
sec.
#no
des
Bes
tnod
eR
SSC
PUse
c.#
node
sB
estn
ode
RSS
5020
194
,662
90.4
113
5,73
10
47,4
8113
9.54
383,
863
383,
310
47,4
81
5020
217
3,33
081
3.68
1,27
7,92
70
99,1
5954
8.20
1,52
1,96
81,
074,
025
99,1
59
5020
310
1,75
734
4.91
549,
877
37,9
7566
,051
670.
381,
843,
030
1,81
3,13
466
,051
5020
470
,347
210.
6132
6,43
90
52,5
8913
2.79
367,
315
294,
049
52,5
89
5020
512
7,48
091
.78
138,
257
061
,223
128.
3435
3,65
830
8,91
861
,223
100
801
339,
592
1,98
7.89
319,
185
295,
947
15,4
153,
600.
003,
122,
527
3,11
9,79
366
,881
100
802
410,
155
0.81
161
095
43,
600.
003,
150,
143
3,15
0,14
210
0,75
6
100
803
376,
343
429.
3892
,775
89,1
5310
,794
2,98
7.66
2,62
2,67
32,
605,
407
10,7
94
100
804
302,
706
0.81
161
096
52,
467.
482,
085,
836
2,08
5,83
696
5
100
805
369,
136
0.96
163
01,
126
3,60
0.00
3,22
9,63
93,
223,
752
371,
680
100
501
611,
976
3,60
0.00
651,
435
012
9,70
63,
600.
003,
284,
774
075
8,26
1
100
502
461,
085
3,60
0.00
650,
955
40,5
5910
3,31
13,
600.
003,
294,
914
063
0,96
6
100
503
443,
940
3,60
0.00
656,
449
013
1,32
73,
600.
003,
399,
950
067
8,23
4
100
504
379,
711
3,60
0.00
664,
523
6311
1,73
83,
600.
003,
345,
433
077
6,80
9
100
505
538,
908
3,60
0.00
652,
547
19,1
6110
3,55
03,
600.
003,
295,
369
061
8,52
0
Algorithm for cardinality-constrained quadratic optimization
Tabl
e7
(Con
tinu
ed)
dK
vFo
rwar
dB
nBC
plex
MIQ
P
RSS
CPU
sec.
#no
des
Bes
tnod
eR
SSC
PUse
c.#
node
sB
estn
ode
RSS
100
201
918,
579
3,60
0.00
2,20
6,24
725
,385
690,
532
3,60
0.00
3,16
7,07
10
758,
261
100
202
807,
784
3,60
0.00
2,04
4,35
30
591,
700
3,60
0.00
3,17
4,39
70
630,
966
100
203
758,
718
3,60
0.00
2,06
9,31
70
622,
728
3,60
0.00
3,22
2,56
80
678,
234
100
204
793,
692
3,60
0.00
2,15
8,69
90
713,
576
3,60
0.00
3,20
9,02
30
776,
809
100
205
796,
714
3,60
0.00
2,29
3,54
70
564,
578
3,60
0.00
3,19
0,98
40
618,
520
500
400
111
,307
,900
3,60
0.00
26,6
872,
507
38,7
813,
600.
0010
2,68
910
2,68
862
,104
,041
500
400
211
,811
,700
3,60
0.00
1,67
50
124,
883
3,60
0.00
58,4
5558
,354
77,8
80,0
26
500
400
315
,760
,100
3,60
0.00
4,32
90
73,6
673,
600.
0049
,378
49,3
0679
,180
,639
500
400
411
,463
,100
3,60
0.00
30
4,64
03,
600.
0051
,088
51,0
8158
,845
,161
500
400
59,
227,
680
3,60
0.00
4,40
30
73,1
953,
600.
0050
,630
50,6
2854
,375
,334
500
100
121
,815
,400
3,60
0.00
50,1
650
15,2
67,3
003,
600.
0055
,981
064
,022
,341
500
100
222
,578
,200
3,60
0.00
51,5
430
14,3
31,1
003,
600.
0013
2,03
313
1,00
072
,052
,226
500
100
325
,562
,500
3,60
0.00
49,7
630
17,8
02,3
003,
600.
0099
,900
99,8
0474
,045
,239
500
100
421
,751
,200
3,60
0.00
52,1
610
15,5
91,1
003,
600.
0012
2,68
90
58,8
63,3
61
500
100
519
,525
,800
3,60
0.00
54,5
510
14,6
81,0
003,
600.
0011
5,09
90
54,4
04,7
34
500
201
34,0
13,8
003,
600.
0010
5,51
770
,717
32,6
26,3
003,
600.
0011
7,45
90
64,0
22,3
41
500
202
36,1
03,3
003,
600.
0014
9,00
946
,825
33,1
20,7
003,
600.
0015
1,84
062
,000
100,
628,
626
500
203
42,8
41,9
003,
600.
0012
2,81
311
2,44
341
,534
,300
3,60
0.00
151,
242
57,0
0095
,185
,439
500
204
36,6
22,5
003,
600.
0010
7,37
581
135
,780
,100
3,60
0.00
121,
559
058
,863
,361
500
205
34,1
51,1
003,
600.
0010
7,39
577
,783
33,8
02,9
003,
600.
0011
4,88
20
54, 4
04,7
34
D. Bertsimas, R. Shioda
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