Multi-Quadratic Integer Programming: Models and Applications

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Multi-Quadratic Integer Programming: Models and Applications

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W. A RT C HAOVALITWONGSE , X IAOZHENG H E , A NTHONY C HEN3 1 Department of Industrial and Systems Engineering, Rutgers University, Piscataway, USA 2 Department of Civil Engineering, University of Minnesota, CE2 , USA 3 Department of Civil and Environmental Engineering, Utah State University, CE2 , USA MSC2000:

CE3

Article Outline Keywords and Phrases Introduction Multi-Quadratic Integer Program Applications Bilinear Problem Minimax Problem Mixed Integer Problem

Solution Techniques Linear Forms of MQIP References

Keywords and Phrases Quadratic programming; Quadratic constraints; Mixed-integer program; Linearization Introduction In this contribution, we consider Multi-Quadratic Programming (MQP) problems, where the objective function is a quadratic function and the feasible region is defined by a finite set of quadratic and linear constraints. They can be formulated as follows: min x T Qx + c T x s.t. x T Aj x + Bj x
bj ; j = 1; : : : ; m

a problem. Indeed, linear mixed 0–1, fractional, bilinear, bilevel, generalized linear complementarity, and many more programming problems are or can easily be reformulated as special cases of MQP. However, there are theoretical and practical difficulties in the process of solving such problems. However, very large linear models can be solved efficiently; whereas MQP problems are in general NP -hard and numerically intractable. The problem of finding a feasible solution is also NP -hard. This is because MQP is a generalization of the linear complementarity problem [28]. The nonlinear constraints in MQP define a feasible region which is in general neither convex nor connected. Moreover, even if the feasible region is a polyhedron, optimizing the quadratic objective function is strongly NP -hard as the resulting problem is considered to be the disjoint bilinear programming problem. Therefore, finding a finite and exact algorithm that solves large MQP problems is impractical. Even for the convex case (when Q and Aj are positive semidefinite), there are very few algorithms for solving MQP problems. However, the MQP constitutes an important part of mathematical programming problems, arising in various practical applications including facility location, production planning, VLSI chip design, optimal design of water distribution networks, and most problems in chemical engineering design. The MQP was first introduced in the seminal paper of Kuhn and Tucker [30]. Later on, the case of MQP with a single quadratic constraint in the problem was discussed in [54,56]. The first general approach for solving MQP problems was proposed in [11], where the following two Lagrange functions for MQP are considered: L1 (x;
) = x T Qx + c T x +

m X


j (x T Aj x  Bj x  bj ) ;

x  0;

where Aj is an (n  n) matrix corresponding to the mth quadratic constraint, and Bj is the jth row of the (m  n) matrix B. MQP plays an important role in modeling many diverse problems. The MQP encompasses many other optimization problems since it provides a much improved model compared to the simpler linear relaxation of Please note that pagination and layout are not final. CE2 Please provide city. CE3 Please provide MSC number.

t c 01

where
and  are the multipliers for the quadratic and bound constraints respectively. A cutting plane algorithm was applied to solve this problem; that is, the algorithm solves a sequence of linear master problems that minimize a piecewise linear function constructed from the Lagrange functions for constant x, and a primal problem with either an unconstrained quadratic function (using L2 (x;
; )) or a quadratic function over the nonnegative orthant (using L1 (x;
)) [20].

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Multi-Quadratic Integer Programming: Models and Applications

Multi-Quadratic Integer Program

Minimax Problem

In this contribution we consider a multi-quadratic integer programming (MQIP) problem with bilevel variables. This problem is a more specific case of MQP. Recently, multi-quadratic zero-one programming problems were proved equivalent to mixed-integer programming problems [15]. In that work, a quadratic zeroone programming was initially proved equivalent to a mixed integer programming problem. Then, the result was extended to the case multi-quadratic programming case. Throughout this paper, we consider a multi-quadratic zero-one programming problem, which has following form:

A related class of global optimization problems are minimax location problems [41], which also lead to quadratic constraints. Production planning and portfolio optimization are examples where so-called chance constrained linear programs occur. These are problems, looking similar to linear programs. However, the matrix describing the linear constraints of such problems is not deterministic, it is a stochastic one. Under certain restrictive assumptions it is possible to transform these stochastic constraints to deterministic quadratic constraints [41], such that in general a problem of type MQP is obtained. In [7] it is shown that nonconvex MQP problems can be used for the examination of special instances of nonlinear bilevel programming problems. Other applications of MQP include the fuel mixture problem encountered in the oil industry [42] and also placement and layout problems in integrated circuit design [8,9].

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Mixed Integer Problem

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P1 :: min f (x) = x T Ax ; s.t. Bx  b ; x T C x  ˛ ; 95

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x 2 f0; 1gn ; ˛ is a constant : Notice x T C x  ˛ essentially represent the same the quadratic constraints as x T Aj x + Bj x
bj in problem (1), due to the binary variables’ property xi xi = xi . Applications Bilinear Problem Each n-dimensional MQP problem can be easily transformed to a 2n-dimensional bilinear problem. A strategy for reducing the necessary dimension of the resulting bilinear program is also proposed [6,27]. However, on the other hand, bilinear optimization problems are nothing else but a special instance of MQP. Pooling problems in petrochemistry, the modular design problem introduced in [16], in particular the multiple modular design problem [6,17] or the more general modularization of product sub-assemblies [45], and special classes of structured stochastic games [19] are only some examples of the wide range of applications of bilinear programming problems. Another large class of optimization problems are problems with linear or quadratic functions additionally involving Boolean variables (i. e., variables xi 2 R with the constraint xi 2 f0; 1g). Another widely explored problem is the problem of packing n 2 N equal circles in a square, which can be transformed to a MQP problem. One looks for the maximum radius r of n non-overlapping circles contained in the unit square. This problem is equivalent to a MQP problem with a linear objective function and concave quadratic constraints.

As described in the previous section, MQP problem can be easily linearized to a mixed integer zeroone problem with the same problem size. In theory and practice, the linearization technique proposed in [15] has been shown to be superior than other conventional linearization techniques. In medical applications, multi-quadratic zero-one problems were used to model epileptic brains for electrode selection problems. Basically, multi-quadratic zero-one problems were solved to identify the location (electrode) sites of the brain that can detect seizure pre-cursors (predict seizures) [29,33,35]. In order to operate in real time, multi-quadratic zero-one problems were linearized to a mixed integer zero-one problem, which is much faster to solve in practice. Hence there are many applications of MQP. Whether the MQP is in practice applicable for solving, for example, problems resulting from integer programming problems, depends on the numerical efficiency of the solution method that is used. Up to now only few methods for solving the considered general case of MQP were proposed in the literature. Most of them result from methods being developed for other more general problem classes. In the next section we will discuss some of the solution techniques.

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Solution Techniques There are many different techniques proposed for solving this type of problems, most of them are of branch and bound type or some type of linearization techniques [4,24,25,26,36,37,38,57]. A disadvantage of the standard linearization technique is the additional variables for each product xi xj , in which the number of new variables is O(n2 ), where n is the number of initial 0–1 variables [4,24,25,57]. The method proposed in [15] needs only O(k n) additional continuous variables, where k is the number of quadratic constraints, and the number of initial 0–1 variables remains the same. A branch-and-bound algorithm for solving MQP problems (and other more general problems), when the objective function is separable and the constraint set is linear, was introduced in [18]. The method evolves solving bounding convex envelope approximating problems over successive partitions of the feasible region. This method was later extended to deal with nonconvex constraints but it generates a number of infeasible solutions and does not, in general, converge in a finite number of iterations [52]. An algorithm for the solution to linear problems with an additional reverse convex constraint was proposed in [14]. The algorithm involves partitioning the feasible region into subsets contained in cones originating at an infeasible vertex of the polytope formed by the linear constraints while ensuring that an interior point of the feasible region is contained in each partition. Later on, an algorithm for the solution to problems with concave objective functions and separable quadratic constraint was proposed in [7]. The algorithm uses piecewise linear approximation for the quadratic constraints and solves a MQP problem as a mixed 0–1 linear problem. This algorithm is similar to the solution approaches for concave quadratic problems [39] and for indefinite quadratic problems [34]. During the last decade, several authors are interested in some special cases of MQP. Also, many extensions of MQP have been discussed in the literature. The problem of minimizing an indefinite quadratic objective subject to two-sided indefinite quadratic constraints was discussed in [53]. Under suitable assumptions, they derived necessary and sufficient optimality conditions and gave some conditions for the existence of solutions for this nonconvex program. While several methods have been suggested for solving MQP problems, numerical solutions of the general problem are still

rarely available in the literature. By using a double duality argument, under suitable assumptions, the MQP is proved to be equivalent to a convex program [55]. In addition, a problem with a concave quadratic function is proved to be equivalent to a minimax convex problem, and thus can be solved in polynomial time via interiorpoints methods. The property is no longer true when Q is an indefinite quadratic function [55].

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Linear Forms of MQIP

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As aforementioned, MQP problems have a close relationship with mixed integer zero-one problem by applying linearization schemes, which have been explored for decades. Although the existing linearization schemes originally were developed for QP instead of MQP, they could be easily applied to MQP, since the quadratic constraints in MQP could be reformulated by using the same technique in linearization considered for the quadratic objective. This section will provide a brief view of major linearization schemes and their applications on MQP problems. No matter what specific reformulation of the linearization schemes, the ideas are the same as replacing the quadratic product xi xj by additional variables. Currently existing linearization schemes were developed in four phases. The prototype of linearization technique arose in 1960s, proposed by Watters [57] and Zangwill [58] (see also Fortet [21,22]). This approach introduces additional binary variables wij for replacement of the products xi xj and additional constraints, xi + xj  wij
1 and xi + xj  2wij ; 8i ; j ; for a guarantee of correct replacement. Taken this approach, the MQP P1 is transformed as following form: X MIP1 :: min f (x; w) = ai i xi

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Multi-Quadratic Integer Programming: Models and Applications

been similarly replaced by additional binary variables wij , and the formulation is consistent with original P1 by additional constraints, xi + xj  wij
1 and xi + xj  2wij ; 8i ; j . Following this seminal work, Glover and Woolsey [24] provided more concise zeroone linear programming formulations, where reformulation rules are given under difference conditions to reduce the numbers of additional constraints and additional variables. In the second phase development of linearization techniques, researchers recognized the additional binary variables wij in MIP1 could be relaxed by continuous ones. Such linearization schemes include the models developed in Glover [23], Glover and Woolsey [25], and Rhys [44]. One scheme with close relationship of the linearization prototype was provided in Glover and Woolsey [25], which introduces additional cut constraints xi  wij and xj  wij ; 8i ; j enforcing the additional continuous variables wij to be binary. However, this technique doubles the number of additional constraints added and thereafter enlarges the size of original MQP problems. A straightforward generalization of xi  xi xj ; 8j generated their further improvement of this technique in [25] that used alternative conP cise constraints (n  i )xj  (j >i ) wij ; 8i to enforce the additional variables to be binary, with somewhat fewer constraints. Applying such linearization technique, the MQP P1 has the following representation: X MIP2 :: min f (x; w) = ai i xi i

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˛ is a constant and A = (aij ); C = (cij ) : 281 282 283 284 285 286 287 288

The main difference between MIP1 and MIP2 is the continuity of wij and the smaller number of additional constraints. Beyond the linearization technique in [25], Glover first noticed Petersen’s work [40], where the cross products in the model are considered by their upper and lower bounds. Following this idea, Glover, in [23], firstly proposed a linearization technique introducing dif-

ferent continuous variables wi to the pioneer research. In his linearization scheme, the additional continuP ous variables are defined by wi = xi j aij xj , where xi are binary variables in original model and aij are quadratic coefficients in the objective function. FurP ther define the lower and upper bounds of j aij xj P P + by A
j faij jaij < 0g and Ai = j faij jaij > 0g, i = respectively. Taking the cross products and binary variables into consideration, the additional inequalities P
A+i xi  wi  A
j aij xj  Ai (1  xi )  i xi and P + wi  j aij xj  Ai (1  xi ) provide the equivalence of original QP model. Applied such linearization technique, the MQP P1 has a different structure as follows: X MIP3 :: min f (w) = wi ; s.t. Bx  b; i

A+i xi

 wi 

X

Ci+ xi

aij xj 

Ci
xi ; 8i ;

 xi )8i

X

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aij xj  A
i (1  xi )

j

 vi 

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j

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(1

 xi )

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j

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j

˛ is a constant : Notice, in MIP3 , the quadratic constraint x T C x  ˛ is P replaced by a series of inequalities i vi  ˛; Ci+ xi  P
vi  Ci
xi ; 8i ; j cij xj  Ci (1  xi )  vi  P + xi )8i , which follow the definitions j cij xj  Ci (1 P P in [23] as: vi = xi j cij xj , Ci
= j fcij jcij < 0g, P and Ci+ = j fcij jcij > 0g. Compared MIP3 with MIP1 and MIP2 , the most important improvement of this linearization technique is that the numbers of additional variables and constraints reduce from O(n2 ) to O(n). Some recent papers [1,2] proposed furtherimproved linearization techniques based on the strategy of Glover’s technique, either providing concise formulation or generating tighter upper/lower bounds. The linearization techniques in the third phase development considered the transformation from the direction of tightness instead of problem size. One typical technique included in this category is the famous Sherali–Adams Reformulation-Linearization Technique (as RLT in short) [3,4,5,49,50,51] which provides widerange applications. The development milestones of RLT can be found in a recent memorial paper written by Sherali [46]. Interested readers could follow this paper

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to find the development details of the linearization scheme. Some practical applications of linearization technique in early 1980s (e. g. [12] considered for solving notorious quadratic assignment problem) generated the experiences that the linearization techniques are practically inefficient although they may have small problem size. Such experience intrigued some researchers to provide better LP structures with tighter bounds, which offer better computational efficiencies, rather than to pursue smaller problem size. The linearization technique shown in [3] provides a structure having tighter bounds for zero-one QP. The transformation happens not only replacing the cross products xi xj in the model but also reconstructing the constraints to obtain the tightness. The example given in [3] not only includes the additional constraints and continuous variables, but reconstructs the linear constraints by multiplying xj and 1  xj , respectively. Applying this linearization technique to MQP, P1 is transformed as follows: X MIP4 :: min f (x; w) = ai i xi i

+

XX X (aij + aj i )wij ; s.t. Bki xi  ˇk i



X i i

8j ; k;

Bki wij + X

X

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i >j

ci i xi +

i

XX i

were also provided between the RLT strategy and the linearization techniques in Watters [57], Zangwill [58], Petersen [40], Glover and Woolsey [24,25], and Glover [23]. Along with this direction, Sherali and Adams [48,49,50] generated a hierarchy of relaxations for zero-one polynomial problems. This relaxation strategy generalizes the idea in [3] by introducing a select set of d-degree polynomial terms or factors, where d is an integer less than the number of binary variables. Multiplying the feasible set by d-degree polynomial terms, as the authors showed, obtains an equivalent reformulation, for each d = 1; : : : ; n, which can enforce the binary restrictions on the original x variables. And these papers also proved that, when d = n, the resulting linear system characterizes the convex hull of feasible solutions, and therefore is tighter than any other linearization techniques. The most recent development, as the final phase, of linearization technique is proposed by Chaovalitwongse et al. [15]. The authors took the dual variables into account, and proposed a new linearization technique based on KKT optimality conditions. Their approach was originally considered for MQP, and is not hard to be utilized for zero-one QP problems. The transformation of MQP P1 using this linearization strategy can be shown as follows.

(cij + cj i )wij  ˛;

e T z  M 0 e T x  ˛; z
2M 0 x;

8i ; j ; x 2 f0; 1g ; wij  0;

x 2 f0; 1gn; yi ; si ; zi  0;

where ˛ is a constant and A = (aij ); C = (cij );

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Notice the linear constraints Bx  b are reconstructed as by multiplying xj and 1  xj and then have much more complicated but tighter representations. The authors also provided the rigorous proof that the construction is tighter than the linearizaiton provided by Glover [23]. Other than that, this formulation uses the inequalities xi  wij and xj  wij instead of P (n  i )xj  (j >i ) wij which will weaken the model’s tightness as pointed out by the authors. Using this idea of multiplying xj and 1  xj to the feasible set Adams and Sherali [4] provided a linearization strategy to more general MIP with cross products between continuous and binary variables. Comparisons

where M 0 = kC k1 and M = kAk1 :

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Notice the additional variables including s; y and z, which are introduced from the Lagrangian function of MQP.

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Theorem 1 P1 has an optimal solution x 0 iff there exist y 0 ; s 0 ; z 0 such that (x 0 ; y 0 ; s 0 ; z 0 ) is an optimal solution of MIP5 . Proof 1 See [15].

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To conclude all the linearization schemes shown herein, we provide a table aggregating the numbers of additional variables and constraints for these techniques as a brief comparison. Assuming we have k linear conP straints i Bj i xi  bj ; j = 1; : : : ; k and m quadratic constraints x T Cj x  ˛j ; j = 1; : : : ; m, in an MQP.

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Also assume that the number of binary variables n  k and n  m. Then the number of additional variables and constraints of the linearized forms applying different techniques can be shown in the table as follows: Models

P1

Additional constraints 0

MIP 1 MIP 2 MIP 3 MIP 4 MIP 5 O(n 2 ) O(n 2 ) O(nm) O(n 2 ) O(nm) O(n 2 ) O(n 2 ) O(nm) O(n 2 ) O(nm)

Addiontal variables

0

Total constraints

O(m + k) O(n 2 ) O(n 2 ) O(nm) O(n 2 ) O(nm)

Total variables

O(n)

O(n 2 ) O(n 2 ) O(nm) O(n 2 ) O(nm)

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Notice that the problem size is not the only reason of computational efficiencies. The tightness of linearization schemes, as pointed out by [46], may significantly change the effectiveness of the techniques for MQP. In terms of solution methods, there are many studies in the literature dealing with the MQP. Most of them apply a technique called semidefinite programming to solve the problem. Specifically, these approaches include special branch-and-bound [8,9,31,43], branchand-cut [10], lift-and-project [10], and the state-of-theart Interior Point method [13,32]. Some of them have been applied in the commercial software package, e. g., the solvers BARON and CPLEX in GAMS. References 1. Adams WP, Forrester RJ (2005) A simple recipe for concise mixed 0–1 linearizations. Oper Res Lett 33:55–61 2. Adams WP, Forrester RJ, Glover FW (2004) Comparison and enhancement strategies for linearizing mixed 0–1 quadratic programs. Discret Optim 1:99–120 3. Adams WP, Sherali HD (1986) A tight linearization and an algorithm for zero-one quadratic programming problems. Manag Sci 32:1274–1290 4. Adams WP, Sherali HD (1990) Linearization strategies for a class of zero-one mixed integer programming problems. Oper Res 38(2):217–226 5. Adams WP, Sherali HD (1993) Mixed-integer bilinear programming problems. Math Program 59(3):279–305 6. Al-Khayyal FA (1992) Generalized bilinear programming: Part I. Models, applications and linear programming relaxation. Eur J Oper Res 60:306–314 7. Al-Khayyal FA, Horst R, Pardalos PM (1992) Global optimization of concave functions subject to separable quadratic constraints: An application to bilevel programming. Ann Oper Res 34:125–147 8. Al-Khayyal FA, Larsen C, van Voorhis T (1995) A relaxation method for nonconvex quadratically constrained quadratic programs. J Glob Optim 6:215–230 9. Al-Khayyal FA, van Voorhis T (1996) Accelerating convergence of branch-and-bound algorithms for quadratically constrained optimization problems. In: Floudas CA (ed) State of CE4

the art in global optimization: computational methods and applications. Kluwer Academic Publishers, CE4

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10. Audet C, Hansen P, Jaumard B, Savard G (2000) A branch and cut algorithms for nonconvex quadratically constrained program. Math Program 87:131–152

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11. Baron DP (1972) Quadratic programming with quadratic constraints. Nav Res Logist Q 19:253–260

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12. Bazaraa MS, Sherali HD (1980) Benders’ partitioning applied to a new formulation of the quadratic assignment problem. Nav Res Logist Q 27(1):28–42

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13. Ben-Tal A, Zibulevsky M (1997) Penalty/Barrier multiplier methods for convex programming problems. SIAM J Optim 7(2):347–366

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14. Ben-Saad S (1989) An algorithm for a class of nonlinear convex optimization problems. PhD thesis. University of California, Los Angeles

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15. Chaovalitwongse WA, Pardalos PM, Prokoyev OA (2004) Reduction of Multi-Quadratic 0–1 Programming Problems to Linear Mixed 0–1 Programming Problems. Oper Res Lett 32(6):517–522

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16. Evans DH (1963) Modular design – A special case in nonlinear programming. Oper Res 11:637–647

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18. Falk JE, Soland RM (1969) An algorithm for separable nonvox programming problems. Manag Sci 15(9):550–569

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20. Floudas CA, Visweswaran V (1995) Quadratic optimization. In: Horst R, Pardalos PM (eds) Handbook of Global Optimization. Kluwer Academic Publishers, CE4 , pp 217–269

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26. Hansen P (1979) Methods of nonlinear 0–1 programming. Ann Discret Math 5:53–70

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27. Hansen P, Jaumard B (1992) Reduction of indefinite quadratic programs to bilinear programs. J Glob Optim 2:41–60

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