A Global Optimization Algorithm for Generalized Quadratic Programming

Abstract

We present a global optimization algorithm for solving generalized quadratic programming (GQP), that is, nonconvex quadratic programming with nonconvex quadratic constraints. By utilizing a new linearizing technique, the initial nonconvex programming problem (GQP) is reduced to a sequence of relaxation linear programming problems. To improve the computational efficiency of the algorithm, a range reduction technique is employed in the branch and bound procedure. The proposed algorithm is convergent to the global minimum of the (GQP) by means of the subsequent solutions of a series of relaxation linear programming problems. Finally, numerical results show the robustness and effectiveness of the proposed algorithm.

1. Introduction

In this paper, we consider the following generalized quadratic programming problem: (GQP):min𝑔₀(𝑥)=𝑥^𝑇𝑄₀𝑥+𝑑^𝑇₀𝑥s.t.𝑔_𝑖(𝑥)=𝑥^𝑇𝑄_𝑖𝑥+𝑑^𝑇_𝑖𝑥≤𝑏_𝑖,𝑖=1,…,𝑚,𝑥∈𝑋⁰=𝑥∈𝑅^𝑛∶𝑙⁰≤𝑥≤𝑢⁰, (1) where 𝑄_𝑖∈𝑅^𝑛×𝑛 (𝑖=0,1,…,𝑚) are all symmetric matrices, 𝑑₀,𝑑_𝑖∈𝑅^𝑛, 𝑏_𝑖∈𝑅,  𝑖=1,…,𝑚; 𝑙⁰=(𝑙⁰₁,…,𝑙⁰_𝑛)^𝑇,  𝑢⁰=(𝑢⁰₁,…,𝑢⁰_𝑛)^𝑇.

Generalized quadratic programming problems are worthy of study, and it is not only because they have broad applications in heat exchanger network design engineering, financial management, statistics, optimal controllers, and so on [1, 2], but also because many other nonlinear problems can be transformed into this form [3, 4], and we often approximate nonlinear programming via solution of a series of quadratic programming problems. In addition, from a research point of view, these problems exist as significant theoretical and computational challenges. This is mainly because these problems are global optimization problem; that is, they are known to generally possess multiple local optimal solutions that are not globally optimal. So it is necessary to establish good solution algorithm for generalized quadratic programming problem (GQP).

In the last decades, many solution methods have been developed for globally solving special form and general form of the (GQP). For examples, when constraint condition is a box constraint, a branch-and-cut algorithm [5] has been proposed by Vandenbussche and Nemhauser; when each function 𝑔_𝑖(𝑥) (𝑖=1,…,𝑚) is a linear function, that is, GQP is an indefinite quadratic programming with linear constraints, then two efficient algorithms [6, 7] have been available; when each function 𝑔_𝑖(𝑥) (𝑖=1,…,𝑚) is a convex function, that is, GQP is one nonconvex quadratic programming with convex constraints, then two feasible algorithms [8, 9] have been presented for globally solving the GQP; by utilizing lagrangian underestimates and the interval newton method, Voorhis [10] has proposed a global optimization algorithm for generalized quadratic programming; by applying a reformulation linearization technique, Sherali and Tuncbilek [11] have presented a feasible method for polynomial programming problems. In addition, Qu et al. [12] Shen and Liu [13] have proposed two different branch and bound algorithms for generalized quadratic programming problem using linearizing method, respectively.

In this paper, we will present a global optimization algorithm for solving the (GQP) by combining branch and bound operation with a range reduction technique. The main features of our algorithm are as follows. (1) A new linearizing technique is presented; utilize the technique to systematically convert the GQP problem into a series of relaxation linear programming problems, and the solutions of these converted problems can infinitely approximate the global optimum of the original GQP problem by a successive refinement process. (2) The proposed linearization techniques for generating relaxation linear programming problems are embedded within a branch and bound scheme without increasing new variables and constraints. (3) The proposed relaxation linear programming for the GQP can be easily solved, which is more convenient in computation than the relaxation convex programming; thus, any effective linear programming algorithm can be used to solve this problem. (4) To accelerate the convergent speed of the proposed algorithm, a range reduction technique is applied in the proposed branch and bound procedure, which provides a theoretical possibility to delete a large part of the currently investigated region where there exists no global optimal solution of the (GQP). (5) Combing the new linearizing technique, branch and bound scheme, and range reduction technique, a branch-reduction-bound algorithm is presented. Finally, numerical results show that the proposed method can globally solve all test problems in obtaining global optimal solutions within a given tolerance error.

The remainder of this paper is organized as follows. The next section describes the linearizing method and the relaxation linear programming of the original problem is established. Section 3 states the branch-reduction-bound algorithm and its convergence. In Section 4 numerical experimental results are given. Finally, the conclusions are presented.

2. Relaxation Linear Programming

The principal structure in the development of a solution procedure for solving the (GQP) is the construction of lower bounds for this problem, as well as for its partitioned subproblems. A lower bound on the solution of the (GQP) and its partitioned subproblems can be obtained by solving a relaxation linear programming of the (GQP). In order to generate the relaxation linear programming, the proposed strategy is that underestimates each quadratic function 𝑔_𝑖(𝑥) (𝑖=0,1,…,𝑚) with a linear function.

In the following, we assume that 𝑋^𝑘={𝑥∈𝑅^𝑛∣𝑙^𝑘≤𝑥≤𝑢^𝑘}⊆𝑋⁰ with 𝑙^𝑘=(𝑙^𝑘₁,…,𝑙^𝑘_𝑛)^𝑇, 𝑢^𝑘=(𝑢^𝑘₁,…,𝑢^𝑘_𝑛)^𝑇. Let 𝜆_𝑖 be the minimum eigenvalues of the matrix 𝑄_𝑖. Select 𝜃_𝑖 such that 𝜃_𝑖=0,if𝜆_𝑖𝜆≥0,_𝑖+𝜌,if𝜆_𝑖<0, (2)where 𝜌 is a small positive number.

For convenience, for any 𝑥∈𝑋^𝑘, for each 𝑖=0,1,…,𝑚, define the function 𝑔^𝐿_𝑖(𝑥)=𝑑_𝑖+2𝑄_𝑖+𝜃_𝑖𝐼^𝑇𝑢^𝑘−2𝜃_𝑖𝑢^𝑘𝑇𝑥+2𝜃_𝑖𝑢^𝑘𝑇𝑙^𝑘−𝜃_𝑖𝑙^𝑘𝑇𝑙^𝑘−𝑢^𝑘𝑇𝑄_𝑖+𝜃_𝑖𝐼𝑢^𝑘. (3)

Theorem 1. For each 𝑖=0,1,…,𝑚, for all 𝑥∈𝑋^𝑘={𝑥∈𝑅^𝑛∣𝑙^𝑘≤𝑥≤𝑢^𝑘}⊆𝑋⁰, by the definition of 𝑔_𝑖(𝑥) and 𝑔^𝐿_𝑖(𝑥), we have the following conclusions:(i)𝑔^𝐿_𝑖(𝑥)≤𝑔_𝑖(𝑥); (ii)‖𝑔_𝑖(𝑥)−𝑔^𝐿_𝑖(𝑥)‖→0 as ‖𝑢^𝑘−𝑙^𝑘‖→0.

Proof. (1) For each 𝑗=1,…,𝑛, by the mean value theorem, for the function 𝑥²_𝑗 over the interval [𝑙^𝑘_𝑗,𝑢^𝑘_𝑗], we have 𝑥²_𝑗=2𝜉^𝑘_𝑗𝑥_𝑗−𝑙^𝑘_𝑗+𝑙^𝑘_𝑗²≤2𝑢^𝑘_𝑗𝑥_𝑗−𝑙^𝑘_𝑗+𝑙^𝑘_𝑗²=2𝑢^𝑘_𝑗𝑥_𝑗−2𝑢^𝑘_𝑗𝑙^𝑘_𝑗+𝑙^𝑘_𝑗², (4) where 𝜉^𝑘_𝑗=𝛼𝑙^𝑘_𝑗+(1−𝛼)𝑢^𝑘_𝑗,𝛼∈[0,1].
From the definition of the function 𝑔^𝐿_𝑖(𝑥) and the equivalent deformation of the function 𝑔_𝑖(𝑥), we can get that 𝑔_𝑖(𝑥)=𝑥^𝑇𝑄_𝑖𝑥+𝑑^𝑇_𝑖𝑥=𝑢^𝑘−𝑥^𝑇𝑄_𝑖+𝜃_𝑖𝐼𝑢^𝑘−𝑥+𝑑^𝑇_𝑖𝑥−𝜃_𝑖^𝑛_𝑗=1𝑥²_𝑗+2𝑢^𝑘𝑇×𝑄_𝑖+𝜃_𝑖𝐼𝑥−𝑢^𝑘𝑇𝑄_𝑖+𝜃_𝑖𝐼𝑢^𝑘≥𝑑^𝑇_𝑖𝑥−𝜃_𝑖^𝑛_𝑗=1𝑥²_𝑗+2𝑢^𝑘𝑇×𝑄_𝑖+𝜃_𝑖𝐼𝑥−𝑢^𝑘𝑇𝑄_𝑖+𝜃_𝑖𝐼𝑢^𝑘≥𝑑^𝑇_𝑖𝑥+𝜃_𝑖^𝑛_𝑗=1−2𝑢^𝑘_𝑗𝑥_𝑗+2𝑢^𝑘_𝑗𝑙^𝑘_𝑗−𝑙^𝑘_𝑗²+2𝑢^𝑘𝑇𝑄_𝑖+𝜃_𝑖𝐼𝑥−𝑢^𝑘𝑇𝑄_𝑖+𝜃_𝑖𝐼𝑢^𝑘=𝑑_𝑖+2𝑄_𝑖+𝜃_𝑖𝐼^𝑇𝑢^𝑘−2𝜃_𝑖𝑢^𝑘𝑇𝑥+2𝜃_𝑖𝑢^𝑘𝑇𝑙^𝑘−𝜃_𝑖𝑙^𝑘𝑇𝑙^𝑘−𝑢^𝑘𝑇𝑄_𝑖+𝜃_𝑖𝐼𝑢^𝑘=𝑔^𝐿_𝑖(𝑥). (5) That is, 𝑔^𝐿_𝑖(𝑥)≤𝑔_𝑖(𝑥),𝑖=0,1,…,𝑚. (6)
(2) By the definition of the function 𝑔^𝐿_𝑖(𝑥) and the proof of conclusion (i), we have 𝑔_𝑖(𝑥)−𝑔^𝐿_𝑖(𝑥=𝑥_𝑇)𝑄_𝑖𝑥+𝑑^𝑇_𝑖𝑥−𝑑_𝑖+2𝑄_𝑖+𝜃_𝑖𝐼^𝑇𝑢^𝑘−2𝜃_𝑖𝑢^𝑘𝑇𝑥+2𝜃_𝑖𝑢^𝑘𝑇𝑙^𝑘−𝜃_𝑖𝑙^𝑘𝑇𝑙^𝑘−𝑢^𝑘𝑇𝑄_𝑖+𝜃_𝑖𝐼𝑢^𝑘=𝑢^𝑘−𝑥^𝑇𝑄_𝑖+𝜃_𝑖𝐼𝑢^𝑘−𝑥+𝑑^𝑇_𝑖𝑥−𝜃_𝑖^𝑛_𝑗=1𝑥²_𝑗+2𝑢^𝑘𝑇×𝑄_𝑖+𝜃_𝑖𝐼𝑥−𝑢^𝑘𝑇𝑄_𝑖+𝜃_𝑖𝐼𝑢^𝑘−𝑑_𝑖+2𝑄_𝑖+𝜃_𝑖𝐼^𝑇𝑢^𝑘−2𝜃_𝑖𝑢^𝑘𝑇𝑥+2𝜃_𝑖𝑢^𝑘𝑇𝑙^𝑘−𝜃_𝑖𝑙^𝑘𝑇𝑙^𝑘−𝑢^𝑘𝑇𝑄_𝑖+𝜃_𝑖𝐼𝑢^𝑘=𝑢^𝑘−𝑥^𝑇𝑄_𝑖+𝜃_𝑖𝐼𝑢^𝑘−𝑥+𝜃_𝑖^𝑛_𝑗=12𝑢^𝑘_𝑗𝑥_𝑗−2𝑢^𝑘_𝑗𝑙^𝑘_𝑗+𝑙^𝑘_𝑗²−𝑥²_𝑗=𝑢^𝑘−𝑥^𝑇𝑄_𝑖+𝜃_𝑖𝐼𝑢^𝑘−𝑥+𝜃_𝑖^𝑛_𝑗=1𝑥_𝑗−𝑙^𝑘_𝑗𝑢^𝑘_𝑗−𝑥_𝑗+𝑢^𝑘_𝑗−𝑙^𝑘_𝑗≤𝑢^𝑘−𝑥^𝑇𝑄_𝑖+𝜃_𝑖𝐼𝑢^𝑘−𝑥+𝜃_𝑖^𝑛_𝑗=1𝑥_𝑗−𝑙^𝑘_𝑗𝑢^𝑘_𝑗−𝑥_𝑗+𝑢^𝑘_𝑗−𝑙^𝑘_𝑗≤𝑢^𝑘−𝑥^𝑇𝑄_𝑖+𝜃_𝑖𝐼𝑢^𝑘−𝑥+𝜃_𝑖𝑥−𝑙^𝑘𝑇𝑢^𝑘−𝑥+𝜃_𝑖𝑥−𝑙^𝑘𝑇𝑢^𝑘−𝑙^𝑘≤𝑢^𝑘−𝑙^𝑘𝑇𝑄_𝑖+𝜃_𝑖𝐼𝑢^𝑘−𝑙^𝑘+2𝜃_𝑖𝑢^𝑘−𝑙^𝑘𝑇𝑢^𝑘−𝑙^𝑘=𝑄_𝑖+𝜃_𝑖𝐼𝑢^𝑘−𝑙^𝑘2+2𝜃_𝑖𝑢^𝑘−𝑙^𝑘2. (7) Since 𝑄_𝑖+𝜃_𝑖𝐼𝑢^𝑘−𝑙^𝑘2+2𝜃_𝑖𝑢^𝑘−𝑙^𝑘2𝑢⟶0as^𝑘−𝑙^𝑘⟶0, (8) we have 𝑔_𝑖(𝑥)−𝑔^𝐿_𝑖(𝑥)⟶0as𝑢^𝑘−𝑙^𝑘⟶0. (9) The proof is complete.

By Theorem 1, we can construct the corresponding approximation relaxation linear programming (RLP) of the (GQP) in 𝑋^𝑘 as follows:RLP𝑋^𝑘:min𝑔^𝐿₀(𝑥)s.t.𝑔^𝐿_𝑖(𝑥)≤𝑏_𝑖,𝑖=1,…,𝑚,𝑥∈𝑋^𝑘=𝑥∶𝑙^𝑘≤𝑥≤𝑢^𝑘, (10) where 𝑔^𝐿_𝑖(𝑥)=𝑑_𝑖+2𝑄_𝑖+𝜃_𝑖𝐼^𝑇𝑢^𝑘−2𝜃_𝑖𝑢^𝑘𝑇𝑥+2𝜃_𝑖𝑢^𝑘𝑇𝑙^𝑘−𝜃_𝑖𝑙^𝑘𝑇𝑙^𝑘−𝑢^𝑘𝑇𝑄_𝑖+𝜃_𝑖𝐼𝑢^𝑘. (11)

Based on the relaxation linear programming, every feasible point of the (GQP) in subdomain 𝑋^𝑘 is feasible to the (RLP); and the objective function value of the (RLP) is less than or equal to that of (GQP) over the partitioned set 𝑋^𝑘. Thus, the RLP(𝑋^𝑘) provides a valid lower bound for the solution of the GQP(𝑋^𝑘).

3. Algorithm and Its Convergence

In this section, a branch-reduction-bound algorithm is proposed for globally solving the (GQP). To find a global optimal solution, the main computation is to solve a sequence of relaxation linear programming problems over partitioned subsets of 𝑋⁰. Furthermore, in order to improve the computational efficiency of the proposed algorithm, a range reduction technique is employed to enhance the solution procedure.

The proposed algorithm is based on partitioning the set 𝑋⁰ into subhyperrectangles, and each subhyperrectangle is associated with a node of the branch and bound tree, and each node is associated with a relaxation linear programming problem in corresponding subhyperrectangle. Therefore, at any stage 𝑘 of the algorithm, assume that we obtain a collection of active nodes denoted by Δ_𝑘, say, each is associated with a hyperrectangle 𝑋⊆𝑋⁰, for all 𝑋∈Δ_𝑘. For each such node 𝑋, compute a corresponding lower bound LB(𝑋) of the GQP by solving the RLP, so that the lower bound of optimal value for the (GQP) over the 𝑋⁰ at iteration 𝑘 is updated by LB_𝑘=min{LB(𝑋),∀𝑋∈Δ_𝑘}. Until the solution of the (RLP) turns out to be feasible to the (GQP), we renew the upper bound 𝑔 if necessary. Then, the set of the active nodes Δ_𝑘 will satisfy LB(𝑋)<𝑔, for all 𝑋∈Δ_𝑘, for each iteration 𝑘. We now select an active node to partition its associated hyperrectangle into two subhyperrectangles according to the following branching rule, and calculate the lower bounds for each new node as before. Upon fathoming any nonimproving nodes, we get a set of active nodes for the next iteration, and this process is repeated until termination condition is satisfied.

The critical factor in guaranteeing convergence to a global optimum is the choice of a suitable branching rule. In this paper, we will choose a simple and standard bisection rule. This rule is sufficient to ensure convergence since it drives all the intervals to zero for all variables. This branching rule is stated as follows. Consider any node subproblem identified by the rectangle 𝑋^′=[𝑥^′,𝑥^′]⊆𝑋⁰. Let 𝑞=argmax{𝑥^′_𝑖−𝑥^′_𝑖∶𝑖=1,2,…,𝑛}, and partition 𝑋^′ into two subrectangles 𝑋^′₁ and 𝑋^′₂ by bisection the interval [𝑥^′_𝑞,𝑥^′_𝑞] into the subintervals [𝑥^′_𝑞,(𝑥^′_𝑞+𝑥^′_𝑞)/2] and [(𝑥^′_𝑞+𝑥^′_𝑞)/2,𝑥^′_𝑞].

3.1. Range Reduction Technique

To improve computational efficiency of the proposed algorithm, a range reduction technique proposed in [13, 14] is applied in the algorithm, which is reformulated as the following Theorems 2 and 3. In iteration 𝑘, we will check whether or not there exists global optimal solution of the GQP(𝑋⁰) over 𝑋. The proposed technique aims at reducing the rectangle 𝑋 without deleting any global optimal solution of the GQP(𝑋⁰) over 𝑋. For convenience, for all 𝑥∈𝑋={𝑥∈𝑅^𝑛∣𝑙≤𝑥≤𝑢}⊆𝑋⁰ with 𝑙=(𝑙₁,𝑙₂,…,𝑙_𝑛)^𝑇 and 𝑢=(𝑢₁,𝑢₂,…,𝑢_𝑛)^𝑇. Without loss of generality, we rewrite the RLP(𝑋) as the following form: RLP(𝑋):min𝑔^𝐿₀(𝑥),s.t.𝑔^𝐿_𝑖(𝑥)≤𝑏_𝑖{},,𝑖=1,…,𝑚,𝑥∈𝑋=𝑥∶𝑙≤𝑥≤𝑢 (12) where 𝑔^𝐿_𝑖(𝑥)=^𝑛_𝑗=1𝑐_𝑖𝑗𝑥_𝑗+𝛿_𝑖,𝑖=0,1,…,𝑚. (13)

Let 𝑔 be a known upper bound of the optimal value of the GQP(𝑋⁰), and set RLB_𝑖=^𝑛_{𝑗=1,𝑐𝑖𝑗>0}𝑐_𝑖𝑗𝑙_𝑗+^𝑛_{𝑗=1,𝑐𝑖𝑗<0}𝑐_𝑖𝑗𝑢_𝑗+𝛿_𝑖,𝑖=0,1,…,𝑚,𝑅𝑔^𝐿_𝑖𝑝=^𝑛_{𝑗=1,𝑗≠𝑝,𝑐𝑖𝑗>0}𝑐_𝑖𝑗𝑙_𝑗+^𝑛_{𝑗=1,𝑗≠𝑝,𝑐𝑖𝑗<0}𝑐_𝑖𝑗𝑢_𝑗+𝛿_𝑖,𝛾𝑖=0,1,…,𝑚,𝑝=1,…,𝑛,_𝑝=𝑔−𝑅𝑔^𝐿_0𝑝𝑐_0𝑝𝜇,𝑝=1,…,𝑛,_𝑖𝑝=𝑏_𝑖−𝑅𝑔^𝐿_𝑖𝑝𝑐_𝑖𝑝,𝑖=1,…,𝑚,𝑝=1,…,𝑛. (14)

Theorem 2. For any subrectangle 𝑋⊆𝑋⁰, the following conclusions hold.(i)If 𝑅𝐿𝐵₀>𝑔, then there exists no global optimal solution of the GQP(𝑋⁰) over 𝑋.(ii)If 𝑅𝐿𝐵₀≤𝑔, then, for each 𝑝∈{1,2,…,𝑛}, if 𝑐_0𝑝>0, then there does not exist global optimal solution of the GQP(𝑋⁰) over 𝑋=(𝑋_𝑗)_𝑛×1; if 𝑐_0𝑝<0, there is no global optimal solution of the GQP(𝑋⁰) over 𝑋=(𝑋_𝑗)_𝑛×1, where 𝑋_𝑗=𝑋_𝑗𝛾,𝑗≠𝑝,𝑗=1,…,𝑛,_𝑝,𝑢_𝑝𝑋_𝑝𝑋,𝑗=𝑝;_𝑗=𝑋_𝑗𝑙,𝑗≠𝑝,𝑗=1,…,𝑛,_𝑝,𝛾_𝑝𝑋_𝑝,𝑗=𝑝. (15)

Proof. (i) If RLB₀>𝑔, then for all 𝑥∈𝑋, 𝑔^𝐿₀(𝑥)≥RLB₀>𝑔; that is, 𝑔₀(𝑥)≥𝑔^𝐿₀(𝑥)≥RLB₀>𝑔. (16) Therefore, there exists no global optimal solution of the GQP(𝑋⁰) over 𝑋.
(ii) If RLB₀≤𝑔, then, for each 𝑝∈{1,…,𝑛}, if 𝑐_0𝑝>0, for all 𝑥∈𝑋, we have 𝑥_𝑝>𝛾_𝑝; that is, 𝑥_𝑝>𝑔−𝑅𝑔^𝐿_0𝑝𝑐_0𝑝or𝑐_0𝑝𝑥_𝑝>𝑔−𝑅𝑔^𝐿_0𝑝. (17) Therefore, we get that 𝑔^𝐿₀(𝑥)=^𝑛_{𝑗=1,𝑗≠𝑝}𝑐_0𝑗𝑥_𝑗+𝑐_0𝑝𝑥_𝑝+𝛿₀≥^𝑛_{𝑗=1,𝑗≠𝑝,𝑐0𝑗>0}𝑐_0𝑗𝑙_𝑗+^𝑛_{𝑗=1,𝑗≠𝑝,𝑐0𝑗<0}𝑐_𝑖𝑗𝑢_𝑗+𝛿₀+𝑐_0𝑝𝑥_𝑝=𝑅𝑔^𝐿_0𝑝+𝑐_0𝑝𝑥_𝑝>𝑅𝑔^𝐿_0𝑝+𝑔−𝑅𝑔^𝐿_0𝑝=𝑔. (18) Thus, we have 𝑔₀(𝑥)≥𝑔^𝐿₀(𝑥)>𝑔. (19) Therefore, there does not exist global optimal solution of the GQP(𝑋⁰) over 𝑋.
Similarly, if 𝑐_0𝑝<0, there is no global optimal solution of problem GQP(𝑋⁰) over 𝑋.

Theorem 3. For any subrectangle 𝑋⊆𝑋⁰, for all 𝑖=1,…,𝑚, the following conclusions hold.(i)If 𝑅𝐿𝐵_i>𝑏_𝑖 for some 𝑖∈{1,…,𝑚}, then there exists no global optimal solution of the GQP(𝑋⁰) over 𝑋.(ii)If 𝑅𝐿𝐵_i≤𝑏_𝑖 for each 𝑖∈{1,…,𝑚}, then, for each 𝑝∈{1,2,…,𝑛}, if 𝑐_𝑖𝑝>0, then there is no global optimal solution of the GQP(𝑋⁰) over 𝑋=(𝑋_𝑗)_𝑛×1; if 𝑐_𝑖𝑝<0, there does not exist global optimal solution of the GQP(𝑋⁰) over 𝑋=(𝑋_𝑗)_𝑛×1, where 𝑋_𝑗=𝑋_𝑗𝜇,𝑗≠𝑝,𝑗=1,…,𝑛,_𝑖𝑝,𝑢_𝑝𝑋_𝑝𝑋,𝑗=𝑝;_𝑗=𝑋_𝑗𝑙,𝑗≠𝑝,𝑗=1,…,𝑛,_𝑝,𝜇_𝑖𝑝𝑋_𝑝,𝑗=𝑝. (20)

Proof. (i) If RLB_𝑖>𝑏_𝑖 for some 𝑖∈{1,…,𝑚}, then for all 𝑥∈𝑋, we have 𝑔_𝑖(𝑥)≥𝑔^𝐿_𝑖(𝑥)≥RLB_𝑖>𝑏_𝑖. (21) Therefore, there exists no global optimal solution of the GQP(𝑋⁰) over 𝑋.
(ii) If RLB_𝑖≤𝑏_𝑖 for each 𝑖∈{1,…,𝑚}, then, for each 𝑝∈{1,2,…,𝑛}, if 𝑐_𝑖𝑝>0, for all 𝑥∈𝑋, we have 𝑥_𝑝>𝜇_𝑖𝑝; that is, 𝑥_𝑝>𝑏_𝑖−𝑅𝑔^𝐿_𝑖𝑝𝑐_𝑖𝑝or𝑐_𝑖𝑝𝑥_𝑝>𝑏_𝑖−𝑅𝑔^𝐿_𝑖𝑝. (22) Therefore, we get that 𝑔^𝐿_𝑖(𝑥)=^𝑛_{𝑗=1,𝑗≠𝑝}𝑐_𝑖𝑗𝑥_𝑗+𝑐_𝑖𝑝𝑥_𝑝+𝛿_𝑖≥^𝑛_{𝑗=1,𝑗≠𝑝,𝑐𝑖𝑗>0}𝑐_𝑖𝑗𝑙_𝑗+^𝑛_{𝑗=1,𝑗≠𝑝,𝑐𝑖𝑗<0}𝑐_𝑖𝑗𝑢_𝑗+𝛿_𝑖+𝑐_𝑖𝑝𝑥_𝑝=𝑅𝑔^𝐿_𝑖𝑝+𝑐_𝑖𝑝𝑥_𝑝>𝑅𝑔^𝐿_𝑖𝑝+𝑏_𝑖−𝑅𝑔^𝐿_𝑖𝑝=𝑏_𝑖. (23) Thus, we have 𝑔_𝑖(𝑥)≥𝑔^𝐿_𝑖(𝑥)>𝑏_𝑖. (24) Hence, there does not exist global optimal solution of the GQP(𝑋⁰) over 𝑋.
Similarly, if 𝑐_𝑖𝑝<0, there is no global optimal solution of the GQP(𝑋⁰) over 𝑋.

By Theorems 2 and 3, we can give a range reduction technique to reject some regions where there does not exist the global optimal solution of the GQP(𝑋⁰). Let 𝑋=(𝑋_𝑗)_𝑛×1 with 𝑋_𝑗=[𝑙_𝑗,𝑢_𝑗] (𝑗=1,…,𝑛) be any subrectangle of 𝑋⁰.

Range Reduction Technique

Reduction Rule (i). Calculate RLB₀.

If RLB₀>𝑔, then let 𝑋=∅; otherwise, calculate 𝑅𝑔^𝐿_0𝑝 and 𝛾_𝑝 (𝑝=1,…,𝑛).

If 𝑐_0𝑝>0 and 𝛾_𝑝<𝑢_𝑝 for some 𝑝∈{1,…,𝑛}, then let 𝑢_𝑝=𝛾_𝑝 and 𝑋=(𝑋_𝑗)_𝑛×1 with 𝑋_𝑗=[𝑙_𝑗,𝑢_𝑗] (𝑗=1,…,𝑛).

If 𝑐_0𝑝<0 and 𝛾_𝑝>𝑙_𝑝 for some 𝑝∈{1,…,𝑛}, then let 𝑙_𝑝=𝛾_𝑝 and 𝑋=(𝑋_𝑗)_𝑛×1 with 𝑋_𝑗=[𝑙_𝑗,𝑢_𝑗] (𝑗=1,…,𝑛).

Reduction Rule (ii). For each 𝑖=1,…,𝑀, calculate RLB_𝑖.

If RLB_𝑖>𝑏_𝑖 for some 𝑖∈{1,…,𝑀}, let 𝑋=∅; otherwise, calculate 𝑅𝑔^𝐿_𝑖𝑝 and 𝜇_𝑖𝑝 (𝑖=1,…,𝑀,𝑝=1,…,𝑛).

If 𝑐_𝑖𝑝>0 and 𝜇_𝑖𝑝<𝑢_𝑝 for some 𝑖∈{1,…,𝑀} and 𝑝∈{1,…,𝑛}, then let 𝑢_𝑝=𝜇_𝑖𝑝 and 𝑋=(𝑋_𝑗)_𝑛×1 with 𝑋_𝑗=[𝑙_𝑗,𝑢_𝑗] (𝑗=1,…,𝑛).

If 𝑐_0𝑝<0 and 𝜇_𝑖𝑝>𝑙_𝑝 for some 𝑖∈{1,…,𝑀} and 𝑝∈{1,…,𝑛}, then let 𝑙_𝑝=𝜇_𝑖𝑝 and 𝑋=(𝑋_𝑗)_𝑛×1 with 𝑋_𝑗=[𝑙_𝑗,𝑢_𝑗] (𝑗=1,…,𝑛).

3.2. Branch-Reduction-Bound Algorithm

Let LB(𝑋^𝑘) refer to the optimal value of the (RLP) over the subhyperrectangles 𝑋^𝑘 and let 𝑥^𝑘=𝑥(𝑋^𝑘) refer to an element of corresponding argmin. Combining the former relaxation linear programming problem, branching operation, and range reduction technique together, the basic steps of the proposed algorithm for globally solving the (GQP) may be stated as follows.

Algorithm 4. Step 0 (initialization). Let the iteration counter 𝑘=0, the set of all active nodes Δ₀={𝑋⁰}, and the convergence tolerance 𝜖>0. Let the initial upper bound 𝑔=+∞, and the set of feasible points 𝐹=∅.
Solve the RLP(𝑋⁰), obtain optimal solution 𝑥⁰=𝑥(𝑋⁰) and optimal value LB(𝑋⁰) for the RLP(𝑋⁰), and let LB₀=LB(𝑋⁰). If 𝑥⁰ is feasible to the GQP(𝑋⁰), let 𝑔=𝑔₀(𝑥⁰) and 𝐹=𝐹∪{𝑥⁰}. If 𝑔−LB₀≤𝜖, then stop with 𝑥⁰ as the global 𝜖-optimal solution for the GQP(𝑋⁰). Otherwise, proceed to Step 1.
Step 1 (branching). Using the proposed branching rule to partition 𝑋^𝑘 into two new subhyperrectangles, and denote the set of new partition rectangles by 𝑋^𝑘.
Step 2 (reduction). For each subhyperrectangle 𝑋∈𝑋^𝑘, utilize the proposed range reduction technique to reject a part of the region or the whole region 𝑋, and still denote the remaining region by 𝑋, and still denote the remaining partition set by 𝑋^𝑘.
Step 3 (updating the upper bound). If 𝑋^𝑘≠∅, solve the RLP(𝑋) to obtain LB(𝑋) and 𝑥(𝑋) for each 𝑋∈𝑋^𝑘. For each 𝑋∈𝑋^𝑘, if LB(𝑋)>𝑔, set 𝑋^𝑘∶=𝑋^𝑘⧵𝑋; otherwise, select the midpoint 𝑥^mid of 𝑋; if 𝑥^mid is feasible to the GQP(𝑋⁰), then let 𝐹∶=𝐹∪{𝑥^mid}, and if 𝑥(𝑋) is feasible to the GQP(𝑋⁰), then let 𝐹∶=𝐹∪{𝑥(𝑋)}.
If 𝐹≠∅, update the upper bound 𝑔∶=min_𝑥∈𝐹𝑔₀(𝑥), and denote the best known feasible point by 𝑥^∗∶=argmin_𝑥∈𝐹𝑔₀(𝑥).
Step 4 (updating the lower bound). The partition set remaining is now Δ_𝑘∶=(Δ_𝑘⧵𝑋^𝑘)∪𝑋^𝑘, and update the lower bound LB_𝑘∶=inf_{𝑋∈Δ𝑘}LB(𝑋).
Step 5 (convergence checking). Set Δ_𝑘+1=Δ_𝑘⧵{𝑋∶𝑔−LB(𝑋)≤𝜖,𝑋∈Δ_𝑘}. If Δ_𝑘+1=∅, then stop with 𝑔 as the global optimal value of the (GQP) and 𝑥^∗ as the global optimal solution. Otherwise, select an active node 𝑋^𝑘+1 such that 𝑋^𝑘+1=argmin_{𝑋∈Δ𝑘+1}LB(𝑋), 𝑥^𝑘+1∶=𝑥(𝑋^𝑘+1). Set 𝑘∶=𝑘+1, and return to Step 1.

3.3. Convergence Analysis

In this subsection, the global convergence properties of the proposed algorithm are presented. If the algorithm does not terminate finitely, then the branching rule guarantees all the intervals shrinking to a singleton for all the variables. On the other hand, Theorem 1 guarantees that as ‖𝑢^𝑘−𝑙^𝑘‖→0 the relaxation linear programming problem RLP(𝑋^𝑘) approaches GQP(𝑋^𝑘). So it is not surprising that the algorithm is shown to be the convergent to the global optimal solution.

Theorem 5. The proposed algorithm either terminates finitely with the global 𝜖-optimal solution 𝑥^∗ of problem GQP(𝑋⁰) or generates an infinite sequence of iteration such that along any infinite branch of the branch and bound tree, any accumulation point of the sequence {𝑥^𝑘} will be the global solution of problem GQP(𝑋⁰).

Proof. If the algorithm is finite, suppose that it terminates in iteration 𝑘, 𝑘≥0. Upon termination, we have 𝑔−LB_𝑘≤𝜖. (25) By Steps 0 and 3 in Algorithm 4, we can find a feasible solution 𝑥^∗ for the problem GQP(𝑋⁰) such that 𝑔₀(𝑥^∗)=𝑔, which implies 𝑔₀𝑥^∗−LB_𝑘≤𝜖. (26) Set 𝑣 denotes the optimal value of problem GQP(𝑋⁰); then, by computational method of lower bound, we have LB_𝑘≤𝑣. (27) Since 𝑥^∗ is a feasible solution of problem GQP(𝑋⁰), we have 𝑣≤𝑔₀𝑥^∗. (28) By inequalities (26)–(28), we can get that 𝑣≤𝑔₀𝑥^∗≤LB_𝑘+𝜖≤𝑣+𝜖. (29) That is, 𝑣≤𝑔₀𝑥^∗≤𝑣+𝜖. (30) Therefore, 𝑥^∗ is a global 𝜖-optimal solution of the problem GQP(𝑋⁰).
If the algorithm is infinite, by the proposed algorithm, we know that the lower bound sequence {LB_𝑘} is nondecreasing and bounded by the upper bound min_𝑥∈𝐹𝑔₀(𝑥); therefore, there exists limitation such that LB=lim_𝑘→∞LB_𝑘≤min_𝑥∈𝐹𝑔₀(𝑥). (31) Since {𝑥^𝑘}∈𝑋⁰, where 𝑋⁰ is a bounded close set, there exists a convergent subsequence {𝑥^𝛽}⊆{𝑥^𝑘} and assume lim_𝛽→∞𝑥^𝛽=𝑥^∗. Then by the proposed algorithm, there exists a decreasing subsequence {𝑋^𝜎}⊆𝑋^𝛽 where 𝑋^𝛽∈Δ_𝛽 with 𝑥^𝜎∈𝑋^𝜎, LB_𝜎=LB(𝑋^𝜎)=𝑔^𝐿₀(𝑥^𝜎) and lim_𝜎→∞𝑋^𝜎={𝑥^∗}. By Theorem 1 and the continuity of the function 𝑔₀(𝑥), we can follow that lim_𝜎→∞LB_𝜎=lim_𝜎→∞𝑔^𝐿₀𝑥^𝜎=lim_𝜎→∞𝑔₀𝑥^𝜎=𝑔₀𝑥^∗. (32)
Then all remaining is to demonstrate that 𝑥^∗ is feasible to the GQP(𝑋⁰).
Firstly, since 𝑋⁰ is bounded and closed set, obviously we have 𝑥^∗∈𝑋⁰.
Secondly, we will prove that 𝑔_𝑖(𝑥^∗)≤𝑏_𝑖, 𝑖=1,…,𝑚.
By contradiction, assume 𝑔_𝜏(𝑥^∗)>𝑏_𝜏 for some 𝜏∈{1,…,𝑚}. Since 𝑔^𝐿_𝜏(𝑥) is a continuous function and by Theorem 1, the sequence {𝑔^𝐿_𝜏(𝑥^𝜎)} is convergent to 𝑔_𝜏(𝑥^∗); that is, there exists a 𝜎 such that 𝑔^𝐿_𝜏𝑥^𝜎−𝑔_𝜏𝑥^∗<𝑔_𝜏𝑥^∗−𝑏_𝜏forany𝜎>𝜎. (33) Hence, for any 𝜎>𝜎, we have 𝑔^𝐿_𝜏(𝑥^𝜎)>𝑏_𝜏, which shows that the LRP(𝑋^𝜎) is infeasible and violates the assumption that 𝑥^𝜎=𝑥(𝑋^𝜎). This will bring about a contradiction; thus, the proof is competed.

4. Numerical Experiments

To compare the proposed algorithm with the known methods in computational efficiency and solution quality, some test examples in recent literature are implemented on microcomputer, the algorithm procedure is coded in C++, and each relaxation linear programming is solved by using simplex method, and the convergence tolerance 𝜖 is set to 10⁻⁶ in Examples 6–13 and 10⁻⁵ in Example 14, respectively. These test examples and numerical results compared with the known algorithms are given in Tables 1 and 2.

Table 1 

Numerical results for the test Examples 6–13.

Table 2 

Numerical results for the test Examples 6–13.

Example 6 (see [15]). Consider min−𝑥²₁+𝑥₁𝑥₂+𝑥²₂+𝑥₁−2𝑥₂s.t.𝑥₁+𝑥₂≤6,−2𝑥²₁+𝑥²₂+2𝑥₁+𝑥₂≤−4,1≤𝑥₁≤6,1≤𝑥₂≤6. (34)

Example 7 (see [13, 16]). Consider min𝑥²₁+𝑥²₂s.t.0.3𝑥₁𝑥₂≥1,2≤𝑥₁≤5,1≤𝑥₂≤3. (35)

Example 8 (see [13, 16, 17]). Consider min𝑥₁s.t.4𝑥₂−4𝑥²₁≤1,−𝑥₁−𝑥₂≤−1,0.01≤𝑥₁≤15,0.01≤𝑥₂≤15. (36)

Example 9 (see [18]). Consider min6𝑥²₁+4𝑥²₂+5𝑥₁𝑥₂s.t.−6𝑥₁𝑥₂≤−48,0≤𝑥₁≤10,0≤𝑥₂≤10. (37)

Example 10 (see [19]). Consider min−𝑥₁+𝑥₁𝑥₂^0.5−𝑥₂s.t.−6𝑥₁+8𝑥₂≤3,3𝑥₁−𝑥₂≤3,1≤𝑥₁≤1.5,1≤𝑥₂≤1.5. (38)

Example 11 (see [14]). Consider min𝑥₁1s.t.4𝑥₁+12𝑥₂−1𝑥16²¹−1𝑥16²²1≤1,𝑥14²¹+1𝑥14²²−37𝑥₁−37𝑥₂≤−1,1≤𝑥₁≤5.5,1≤𝑥₂≤5.5. (39)

Example 12 (see [14]). Consider min𝑥₁𝑥₂−2𝑥₁+𝑥₂+1s.t.8𝑥²₂−6𝑥₁−16𝑥₂≤−11,−𝑥²₂+3𝑥₁+2𝑥₂≤7,1≤𝑥₁≤2.5,1≤𝑥₂≤2.225. (40)

Example 13 (see [20]). Consider min−4𝑥₂+𝑥₁−1²+𝑥²₂−10𝑥²₃s.t.𝑥²₁+𝑥²₂+𝑥²₃𝑥≤2,₁−2²+𝑥²₂+𝑥²₃√≤2,2−2≤𝑥₁≤√2,0≤𝑥₂,𝑥₃≤√2. (41)

Example 14 (see [21]). Consider min−^𝑛_𝑖=1𝑥²_𝑖s.t.^𝑗_𝑖=1𝑥_𝑖{},𝑥≤𝑗,𝑗∈1,2,…,𝑛_𝑖≥0,𝑖∈{1,2,…,𝑛}. (42)

The numerical results for Example 14 compared with [21] are given in Table 3.

Table 3 

Numerical results for Example 14.

The numerical results for test Examples 6–14 show that the proposed algorithm is robust and can be used to globally solve generalized quadratic programming (GQP).

5. Concluding Remarks

In this paper, a global optimization algorithm based on a new linearizing method and a range reduction technique is presented for solving the (GQP). The relaxation linear programming of the original problem (GQP) is established by underestimating the objective and constraint functions with linear functions. By making full use of the currently known upper bound and relaxation linear programming of the (GQP), a range reduction technique is presented. The algorithm is convergent to the global minimum through the successive refinement of a linear relaxation of the feasible region and/or of the objective function and the subsequent solution of a series of relaxation linear programming problems. Numerical results show effectiveness and robustness of the proposed algorithm.

Acknowledgments

This paper is supported by the National Natural Science Foundation of China (11171094), the Foundation for University Key Teacher by the Ministry of Education of Henan Province (2010GGJS-140), and the Natural Science Foundation of Henan Educational Committee (2010B110010). The authors acknowledged the help of Yingfeng Zhao while preparing the paper.