Complete Solutions to General Box-Constrained Global Optimization Problems

Abstract

This paper presents a global optimization method for solving general nonlinear programming problems subjected to box constraints. Regardless of convexity or nonconvexity, by introducing a differential flow on the dual feasible space, a set of complete solutions to the original problem is obtained, and criteria for global optimality and existence of solutions are given. Our theorems improve and generalize recent known results in the canonical duality theory. Applications to a class of constrained optimal control problems are discussed. Particularly, an analytical form of the optimal control is expressed. Some examples are included to illustrate this new approach.

1. Introduction

In this paper, we consider the following general box constrained nonlinear programming problem (the primal problem (𝒫) in short): (𝒫)∶min𝑃(𝑥)∣𝑥∈𝒳_𝑎, (1.1) where 𝒳_𝑎={𝑥∈ℛ^𝑛∣ℓ^𝑙≤𝑥≤ℓ^𝑢} is a feasible space, ℓ^𝑙,ℓ^𝑢∈ℛ^𝑛 are two given vectors, and 𝑃(𝑥) is twice continuously differentiable in ℛ^𝑛. Here, we discuss the primal problem (𝒫) involving nonconvexity or convexity in the objective function.

Problem (1.1) appears in many applications, such as engineering design, phase transitions, chaotic dynamics, information theory, and network communication [1, 2]. Particularly, if ℓ^𝑙={0} and ℓ^𝑢={1}, the problem leads to one of the fundamental problems in combinatorial optimization, namely, the integer programming problem [3]. By the fact that the feasible space 𝒳_𝑎 is a closed convex subset of ℛ^𝑛, the primal problem has at least one global minimizer. When (𝒫) is a convex programming problem, a global minimizer can be obtained by many well-developed nonlinear optimization methods based on the Karush-Kuhn-Tucker (or simply KKT ) optimality theory [4]. However, for (𝒫) with nonconvexity in the objective function, traditional KKT theory and direct methods can only be used for solving (𝒫) to local optimality. So, our interest will be mainly in the case of 𝑃(𝑥) being nonconvex on 𝒳_𝑎 in this paper. For special cases of minimizing a nonconvex quadratic function subject to box constraints, much effort and progress have been made on locating the global optimal solution based on the canonical duality theory by Gao (see [5–7] for details). As indicated in [8], the key step of the canonical duality theory is to introduce a canonical dual function, but commonly used methods are not guaranteed to construct it since the general form of the objective function given in (1.1). Thus, there has been comparatively little work in global optimality for general cases.

Inspired and motivated by these facts, a differential flow for constructing the canonical dual function is introduced and a new approach to solve the general (especially nonconvex) nonlinear programming problem (𝒫) is investigated in this paper. By means of the canonical dual problem, some conditions in global optimality are deduced, and global and local extrema of the primal problem can be identified. An application to the linear-quadratic optimal control problem with constraints is discussed. These results presented in this paper can be viewed as an extension and an improvement in the canonical duality theory [8–10].

The paper is organized as follows. In Section 2, a differential flow is introduced to present a general form of the canonical dual problem to (𝒫). The relation of this transformation with the classical Lagrangian method is discussed. In Section 3, we present a set of complete solutions to (𝒫) by the way presented in Section 2. The existence of the canonical dual solutions is also given. We give an analytic solution to the box-constrained optimal control problem via canonical dual variables in Section 4. Meanwhile, some examples are used to illustrate our theory.

2. A Differential Flow and Canonical Dual Problem

In the beginning of this paper, we have mentioned that our primal goal is to find the global minimizers to a general (mainly nonconvex) box-constrained optimization problem (𝒫). Due to the assumed nonconvexity of the objective function, the classical Lagrangian 𝐿(𝑥,𝜎) is no longer a saddle function, and the Fenchel-Young inequality leads to only a weak duality relation: min𝑃≥max𝑃^∗. The nonzero value 𝜃=min𝑃(𝑥)−max𝑃^∗(𝜎) is called the duality gap, where possibly, 𝜃=∞. This duality gap shows that the well-developed Fenchel-Moreau-Rockafellar duality theory can be used mainly for solving convex problems. Also, due to the nonconvexity of the objective function, the problem may have multiple local solutions. The identification of a global minimizer has been a fundamentally challenging task in global optimization. In order to eliminate this duality gap inherent in the classical Lagrange duality theory, a so-called canonical duality theory has been developed [2, 9]. The main idea of this new theory is to introduce a canonical dual transformation which may convert some nonconvex and/or nonsmooth primal problems into smooth canonical dual problems without generating any duality gap and deduce some global solutions. The key step in the canonical dual transformation is to choose the (nonlinear) geometrical operator Λ(𝑥). Different forms of Λ(𝑥) may lead to different (but equivalent) canonical dual functions and canonical dual problems. So far, in most literatures related, the canonical dual transformation is discussed and the canonical dual function is formulated in quadratic minimization problems (i.e., the objective function is the quadratic form). However, for the general form of the objective function given in (1.1), in general, it lacks effective strategies to get the canonical dual function (or the canonical dual problem) by commonly used methods. The novelty of this paper is to introduce the differential flow created by differential equation (2.6) to construct the canonical dual function for the problem (𝒫). Lemma 2.5 guarantees the existence of the differential flow; Theorem 2.3 shows that there is no duality gap between the primal problem (𝒫) and its canonical dual problem (𝑃^𝑑) given in (2.7) via the differential flow; Meanwhile, Theorems 3.1–3.4 use the differential flow to present a global minimizer. In addition, the idea to introduce the set 𝒮 of shift parameters is closely following the works by Floudas et al. [11, 12]. In [12], they developed a global optimization method, 𝛼𝐵𝐵, for general twice-differentiable constrained optimizations proposing to utilize some 𝛼 parameter to generate valid convex under estimators for nonconvex terms of generic structure.

The main idea of constructing the differential flow and the canonical dual problem is as follows. For simplicity without generality, we assume that ℓ^𝑢=−ℓ^𝑙=−ℓ^1/2={ℓ_𝑖}, namely, 𝒳_𝑎={𝑥∈ℛ^𝑛∣−ℓ_𝑖≤𝑥_𝑖≤ℓ_𝑖}, where ℓ_𝑖≠0 for all 𝑖.

Let 𝒮 denote the dual feasible space 𝒮=𝜌∈ℛ^𝑛₊∣∇²𝑃(𝑥)+Diag𝜌>0,∀𝑥∈𝒳_𝑎, (2.1) where ℛ^𝑛₊∶={𝜌∈ℛ^𝑛∣𝜌≥0}, and Diag(𝜌)∈ℛ^𝑛×𝑛 is a diagonal matrix with 𝜌_𝑖,𝑖=1,2,…,𝑛, as its diagonal entries.

Lemma 2.1. The dual feasible space 𝒮 is an open convex subset of ℛ^𝑛₊. If 𝜌∈𝒮, then 𝜌∈𝒮 for any 𝜌≥𝜌.

Proof. Notice that 𝑃(𝑥) is twice continuously differentiable in ℛ^𝑛. For any 𝑥∈𝒳_𝑎, the Hessian matrix ∇²𝑃(𝑥) is a symmetrical matrix. We know that for any given 𝑄=𝑄^T∈ℛ^𝑛×𝑛, {𝜌∈ℛ^𝑛₊∣𝑄+Diag(𝜌)>0} is a convex set. By the fact that the intersection of any collection of convex sets is convex, the dual feasible space 𝒮 is an open convex subset of ℛ^𝑛₊. In addition, it follows from the definition of 𝒮 that 𝜌∈𝒮 for any 𝜌≥𝜌. This completes the proof.

Suppose that 𝜌^∗∈𝒮 and a nonzero vector 𝑥^∗∈𝒳_𝑎 satisfy ∇𝑃𝑥^∗+Diag𝜌^∗𝑥^∗=0. (2.2) A differential flow 𝑥(𝜌) is introduced over a relative small neighborhood of 𝜌^∗ such that ∇𝑃𝑥𝜌+Diag𝜌𝑥𝜌=0,𝑥𝜌^∗=𝑥^∗, (2.3) which is equivalent to ∇²𝑃𝑥𝜌∇𝑥𝜌+Diag𝜌∇𝑥𝜌+Diag𝑥𝜌=0,𝑥𝜌^∗=𝑥^∗, (2.4) where ∇𝑥(𝜌) is the Jacobian of 𝑥 and is a matrix whose 𝑖𝑗th entry is equal to the partial derivative 𝜕𝑥_𝑖/𝜕𝜌_𝑗. Here, we hope to preserve invertibility of the matrix ∇²𝑃(𝑥(𝜌))+Diag(𝜌) on the choice of the neighborhood of 𝜌^∗.

Let 𝐻𝜌,𝑥𝜌=−∇²𝑃𝑥𝜌+Diag𝜌⁻¹. (2.5) Then, a differential flow 𝑥(𝜌) can be defined by the following differential system: 𝑑𝑥_𝑖𝜌=𝐻_𝑖1𝜌𝑥₁𝜌𝑑𝜌₁+𝐻_𝑖2𝜌𝑥₂𝜌𝑑𝜌₂+⋯+𝐻_𝑖𝑛𝜌𝑥_𝑛𝜌𝑑𝜌_𝑛,𝑥_𝑖𝜌^∗=𝑥^∗_𝑖,𝑖=1,2,…,𝑛, (2.6) where 𝐻_𝑖𝑗(𝜌) is the 𝑖𝑗th entry of 𝐻(𝜌,𝑥(𝜌)). Based on the Extension theory [13, 14], the solution 𝑥(𝜌) of the differential system (2.6) can be extended to a space in 𝒮. The canonical dual function is defined as 𝑃^𝑑𝜌=𝑃𝑥𝜌+^𝑛_𝑖=1𝜌_𝑖2𝑥²_𝑖𝜌−ℓ_𝑖. (2.7) Thus, the canonical dual problem for our primal problem (𝒫) can be proposed as follows 𝑃^𝑑∶max𝑃^𝑑𝜌=𝑃𝑥𝜌+^𝑛_𝑖=1𝜌_𝑖2𝑥²_𝑖𝜌−ℓ_𝑖∣𝜌∈𝒮. (2.8) In the following, we show that (𝑃^𝑑) is canonically (i.e., with zero duality gap) dual to (𝒫).

Lemma 2.2. Let 𝑥(𝜌) be a given flow defined by (2.6), and 𝑃^𝑑(𝜌) be the corresponding canonical dual function defined by (2.7). For any 𝜌∈𝒮, we have ∇𝑃^𝑑𝜌=12𝑥²₁𝜌−ℓ₁,12𝑥²₂𝜌−ℓ₂1,…,2𝑥²_𝑛𝜌−ℓ_𝑛^T,∇²𝑃^𝑑𝜌=−Diag𝑥𝜌∇²𝑃𝑥𝜌+Diag𝜌⁻¹Diag𝑥𝜌. (2.9)

Proof. Since 𝑃^𝑑(𝜌) is differentiable, for any 𝜌∈𝒮, 𝜕𝑃^𝑑𝜌𝜕𝜌_𝑖=∇𝑃𝑥𝜌^𝑇𝜕𝑥𝜌𝜕𝜌_𝑖+12𝑥²_𝑖𝜌−ℓ_𝑖+^𝑛_𝑗=1𝜌_𝑗𝑥_𝑗𝜌𝜕𝑥_𝑗𝜌𝜕𝜌_𝑖=∇𝑃𝑥𝜌^𝑇𝜕𝑥𝜌𝜕𝜌_𝑖+𝜌^𝑇𝑥𝜌𝜌Diag𝜕𝑥𝜕𝜌_𝑖+12𝑥²_𝑖𝜌−ℓ_𝑖=∇𝑃𝑥𝜌+Diag(𝜌)𝑥𝜌^𝑇𝜕𝑥𝜌𝜕𝜌_𝑖+12𝑥²_𝑖𝜌−ℓ_𝑖. (2.10) It follows from the process (2.2)–(2.6) that 𝜕𝑃^𝑑𝜌𝜕𝜌_𝑖=12𝑥²_𝑖𝜌−ℓ_𝑖. (2.11) From (2.6), we have 𝜕𝑥_𝑖(𝜌)/𝜕𝜌_𝑗=𝐻_𝑖𝑗(𝜌)𝑥_𝑗(𝜌). By (2.11), then 𝜕²𝑃^𝑑𝜌𝜕𝜌_𝑖𝜕𝜌_𝑗=𝑥_𝑖𝜌𝜕𝑥_𝑖𝜌𝜕𝜌_𝑗=𝑥_𝑖𝜌𝐻_𝑖𝑗𝜌𝑥_𝑗𝜌. (2.12) By the definition of 𝐻(𝜌,𝑥(𝜌)), this concludes the proof of Lemma 2.2.

By Lemma 2.2, the canonical dual function 𝑃^𝑑(𝜌) is concave on 𝒮. For a critical point, 𝜌∈𝒮, 𝜌 must be a global maximizer of (𝑃^𝑑), and it can be solved by many well-developed nonlinear programming methods. If 𝜌∈𝒮 and 𝑥(𝜌)∈𝒳_𝑎, we have 𝜕𝑃^𝑑(𝜌)/𝜕𝜌_𝑖=(1/2)[𝑥²_𝑖(𝜌)−ℓ_𝑖]≤0, and for any 𝜌≥𝜌, by negative definiteness of 𝐻(𝑥(𝜌),𝜌), 𝜕(1/2)𝑥²_𝑖𝜌−ℓ_𝑖𝜕𝜌_𝑖=𝑥_𝑖𝜌𝜕𝑥_𝑖𝜌𝜕𝜌_𝑖=𝐻_𝑖𝑖𝜌𝑥²_𝑖𝜌≤0. (2.13) Thus for any 𝜌≥𝜌, 𝑥(𝜌) will stay in 𝒳_𝑎 and 𝑃^𝑑(𝜌)≤𝑃^𝑑(𝜌).

Theorem 2.3. The canonical dual problem (𝑃^𝑑) is perfectly dual to the primal problem (𝒫) in the sense that if 𝜌∈𝒮 is a KKT point of 𝑃^𝑑(𝜌), then the vector 𝑥=𝑥(𝜌) is a KKT point of (𝒫) and 𝑃(𝑥)=𝑃^𝑑𝜌. (2.14)

Proof. By introducing the Lagrange multiplier vector 𝜆∈ℛ^𝑛 to relax the inequality constraint 𝜌≥0 in 𝒮, the Lagrangian function associated with (𝑃^𝑑) becomes 𝐿(𝜌,𝜆)=𝑃^𝑑(𝜌)−𝜆^𝑇𝜌.Then the KKT conditions of (𝑃^𝑑) become ∇𝐿=∇𝑃^𝑑𝜌𝜆−𝜆=0,_𝑇𝜌=0,𝜆≤0,𝜌≥0. (2.15) Notice that ∇𝑃^𝑑(𝜌)=((1/2)[𝑥²₁(𝜌)−ℓ₁],(1/2)[𝑥²₂(𝜌)−ℓ₂],…,(1/2)[𝑥²_𝑛(𝜌)−ℓ_𝑛])^𝑇. It follows from conditions (2.15) that 𝑥(𝜌) satisfies the complementary conditions of (𝒫). By the definition of the flow 𝑥(𝜌), the equation ∇𝑃(𝑥(𝜌))+Diag(𝜌)𝑥(𝜌)=0 holds. This proved that if 𝜌∈𝒮 is a KKT point of 𝑃^𝑑(𝜌), then the vector 𝑥=𝑥(𝜌) defined by (2.6) is a KKT point of the primal problem (𝒫).
In addition, we have 𝑃^𝑑𝜌=𝑃𝑥𝜌+^𝑛_𝑖=1𝜌_𝑖2𝑥²_𝑖𝜌−ℓ_𝑖=𝑃𝑥𝜌=𝑃(𝑥). (2.16) This completes the proof.

Remark 2.4. Theorem 2.3 shows that by using the canonical dual function (2.7), there is no duality gap between the primal problem (𝒫) and its canonical dual (𝑃^𝑑), that is, 𝜃=0. It eliminates this duality gap inherent in the classical Lagrange duality theory and provides necessary conditions for searching global solutions. Actually, we replace 𝒮 in (𝑃^𝑑) with the space 𝒮_#, 𝒮_#∶={𝜌≥0∣det[∇²𝑃(𝑥)+Diag(𝜌)]≠0,forall𝑥∈𝒳_𝑎} in the proof of Theorem 2.3. Moreover, the inequality of det[∇²𝑃(𝑥)+Diag(𝜌)]≠0,forall𝑥∈𝒳_𝑎 in 𝒮_# is essentially not a constraint as indicated in [5].

Due to introduceing a differential flow 𝑥(𝜌), the constrained nonconvex problem can be converted to the canonical (perfect) dual problem, which can be solved by deterministic methods. In view of the process (2.2)–(2.6), the flow 𝑥(𝜌) is based on the KKT (2.2). In other words, we can solve equation (2.2) backwards from 𝜌^∗ to get the backward flow 𝑥(𝜌),𝜌∈𝒮∩{0≤𝜌≤𝜌^∗}. Then, it is of interest to know whether there exists a pair (𝑥^∗,𝜌^∗) satisfying (2.2).

Lemma 2.5. Suppose that ∇𝑃(0)≠0. For the primal problem (𝒫), there exist a point 𝜌^∗∈𝒮 and a nonzero vector 𝑥^∗∈𝒳_𝑎 such that ∇𝑃(𝑥^∗)+Diag(𝜌^∗)𝑥^∗=0.

Proof. Since 𝒳_𝑎 is bounded and 𝑃(𝑥) is twice continuously differentiable in ℛ^𝑛, we can choose a large positive real 𝑀∈ℛ such that ∇²𝑃(𝑥)+Diag(𝑀𝑒)>0,forall𝑥∈𝒳_𝑎 and sup_𝒳𝑎|(∇𝑃(𝑥))_𝑖|<𝑀ℓ_𝑖,𝑖=1,2,…,𝑛 (𝑒∈ℛ^𝑛 is an 𝑛−vector of all ones). Then, it is easy to verify that ∇𝑃(𝑥)+Diag(𝑀𝑒)𝑥<0 at the point 𝑥=−ℓ^1/2, and ∇𝑃(𝑥)+Diag(𝑀𝑒)𝑥>0 at the point 𝑥=ℓ^1/2.
Notice that the function ∇𝑃(𝑥)+Diag(𝑀𝑒)𝑥 is continuous and differentiable in ℛ^𝑛. It follows from differential and integral calculus that there is a nonzero stationary point 𝑥^∗∈𝒳_𝑎 such that ∇𝑃(𝑥^∗)+Diag(𝑀𝑒)𝑥^∗=0. Let 𝜌^∗=𝑀𝑒∈𝒮. Thus, there exist a point 𝜌^∗∈𝒮 and a nonzero vector 𝑥^∗∈𝒳_𝑎 satisfying (2.2). This completes the proof.

Remark 2.6. Actually, Lemma 2.5 gives us some information to search the desired parameter 𝜌^∗. From Lemma 2.5, we only need to choose a large positive real 𝑀∈ℛ such that ∇²𝑃(𝑥)+Diag(𝑀𝑒)>0,forall𝑥∈𝒳_𝑎 and sup_𝒳𝑎|(∇𝑃(𝑥))_𝑖|<𝑀ℓ_𝑖,𝑖=1,2,…,𝑛. Since ∇𝑃(0)≠0, then it follows from (‖∇²𝑃(𝑥)‖/𝑀)<1 uniformly in 𝒳_𝑎 that there is a unique nonzero fixed point 𝑥^∗∈𝒳_𝑎 such that −∇𝑃𝑥^∗𝑀=𝑥^∗ (2.17) which is equivalent to ∇𝑃(𝑥^∗)+Diag(𝑀𝑒)𝑥^∗=0 by Brown fixed-point theorem. In [11, 12], some good algorithms are given to estimate the bounds of ‖∇²𝑃(𝑥)‖. If there is a positive real number 𝐾 such that ‖∇²𝑃(𝑥)‖≤𝐾,forall𝑥∈𝒳_𝑎, then a properly large parameter 𝑀 can be obtained by the inequalities ∇²𝑃(𝑥)𝑀≤𝐾𝑀<1,𝑀>sup_𝒳𝑎(∇𝑃(𝑥))_𝑖ℓ_𝑖,∀𝑖 (2.18) uniformly on 𝒳_𝑎 for us to use Brown fixed-point theorem. We should choose 𝑀>maxsup_𝒳𝑎∇²𝑃(𝑥),sup_𝒳𝑎(∇𝑃(𝑥))_𝑖ℓ_𝑖,∀𝑖. (2.19) Finally, let 𝜌^∗=𝑀𝑒 which is the desired parameter 𝜌^∗ for us. We will discuss to calculate the parameter 𝜌^∗ in detail by the use of the results in [15, 16] with the future works.

Remark 2.7. Moreover, for the proper parameter 𝜌^∗, it is worth investigating how to get the solution 𝑥^∗ of (2.2) inside of 𝒳_𝑎. For this issue, when 𝑃(𝑥) is a polynomial, we may be referred to [17]. There are results in [17] on bounding the zeros of a polynomial. We may consider for a given bounds to determine the parameter by the use of the results in [17] on the relation between the zeros and the coefficients. We will discuss it with the future works as well. However, the KKT conditions are only necessary conditions for local minimizers to satisfy for the nonconvex case of (𝒫). To identify, a global minimizer among all KKT points remains a key task for us to address in the next section.

3. Complete Solutions to Global Optimization Problems

Theorem 3.1. Suppose that 𝜌 is a KKT point of the canonical dual function 𝑃^𝑑(𝜌) and 𝑥=𝑥(𝜌) defined by (2.6). If 𝜌∈𝒮, then 𝜌 is a global maximizer of (𝑃^𝑑) on 𝒮, and 𝑥 is a global minimizer of (𝒫) on 𝒳_𝑎 and 𝑃(𝑥)=min_{𝑥∈𝒳𝑎}𝑃(𝑥)=max_𝜌∈𝒮𝑃^𝑑𝜌=𝑃^𝑑𝜌. (3.1)

Proof. If 𝜌∈𝒮 is a KKT point of (𝑃^𝑑) on 𝒮, by (2.15), 𝑥(𝜌) stays in 𝒳_𝑎, that is, (1/2)(𝑥²_𝑖(𝜌)−ℓ_𝑖)≤0 for all 𝑖. By Lemma 2.1 and 2.2, it is easy to verify that 𝑥(𝜌)∈𝒳_𝑎 and 𝑃^𝑑(𝜌)≥𝑃^𝑑(𝜌) for any 𝜌≥𝜌.
For any given parameter 𝜌, (𝜌≥𝜌), we define the function 𝑓_𝜌(𝑥) as follows: 𝑓_𝜌(𝑥)=𝑃(𝑥)+^𝑛_𝑖=1𝜌_𝑖2𝑥²_𝑖−ℓ_𝑖. (3.2) It is obvious that 𝑃(𝑥)≥𝑓_𝜌(𝑥) for all 𝑥∈𝒳_𝑎. Since 𝑓_𝜌(𝑥) is twice continuously differentiable in ℛ^𝑛, there exists a closed convex region ℰ containing 𝒳_𝑎 such that on ℰ, ∇𝑃𝑥𝜌+^𝑛_𝑖=1𝜌_𝑖2𝑥²_𝑖𝜌−ℓ_𝑖𝑥𝜌𝜌𝑥𝜌∇=∇𝑃+Diag=0,₂𝑃(𝑥)+^𝑛_𝑖=1𝜌_𝑖2𝑥²_𝑖−ℓ_𝑖=∇²𝑃(𝑥)+Diag𝜌>0,∀𝑥∈𝒳_𝑎. (3.3) This implies that 𝑥(𝜌) is the unique global minimizer of 𝑓_𝜌(𝑥) over ℰ. By (2.7), we have 𝑓_𝜌𝑥𝜌=𝑃𝑥𝜌+^𝑛_𝑖=1𝜌_𝑖2𝑥²_𝑖𝜌−ℓ_𝑖=𝑃^𝑑𝜌. (3.4) Thus, for any 𝜌≥𝜌, 𝑃(𝑥)≥𝑓_𝜌(𝑥)≥min_{𝑥∈𝒳𝑎}𝑓_𝜌(𝑥)=𝑓_𝜌𝑥𝜌=𝑃^𝑑𝜌. (3.5) On the other hand, by the fact that the canonical dual function 𝑃^𝑑(𝜌) is concave on 𝒮, 𝜌 must be a global maximizer of (𝑃^𝑑) on 𝒮, and we have max_𝜌∈𝒮𝑃^𝑑𝜌=𝑃^𝑑𝜌=𝑃𝑥𝜌+^𝑛_𝑖=1𝜌_𝑖2𝑥²_𝑖𝜌−ℓ_𝑖=𝑃𝑥𝜌, (3.6) and for all 𝑥∈𝒳_𝑎, 𝑃(𝑥)≥max_𝜌≥𝜌𝑃^𝑑𝜌=𝑃^𝑑𝜌=𝑃(𝑥). (3.7)
Thus, 𝑥 is a global minimizer of (𝒫) on 𝒳_𝑎 and 𝑃(𝑥)=min_{𝑥∈𝒳𝑎}𝑃(𝑥)=max_𝜌∈𝒮𝑃^𝑑𝜌=𝑃^𝑑𝜌. (3.8) This completes the proof.

Remark 3.2. Theorem 3.1 shows that a vector 𝑥=𝑥(𝜌) is a global minimizer of the problem (𝒫) if 𝜌∈𝒮 is a critical solution of (𝑃^𝑑). However, for certain given 𝑃(𝑥), the canonical dual function might have no critical point in 𝒮. For example, the canonical dual solutions could locate on the boundary of 𝒮. In this case, the primal problem (𝒫) may have multiple solutions.

In order to study the existence conditions of the canonical dual solutions, we let 𝜕𝒮 denote the boundary of 𝒮.

Theorem 3.3. Suppose that 𝑃(𝑥) is a given twice continuously differentiable function, 𝒮≠∅ and 𝜕𝒮≠∅. If for any given 𝜌₀∈𝜕𝒮 and 𝜌∈𝒮, lim_𝛼→0+𝑃^𝑑𝜌₀+𝛼𝜌=−∞, (3.9) then the canonical dual problem (𝑃^𝑑) has a critical point 𝜌∈𝒮, and 𝑥(𝜌) is a global minimizer to (𝒫).

Proof. We first show that for any given 𝜌≥0∈ℛ^𝑛,𝜌≠0, lim_𝛼→∞𝑃^𝑑𝛼𝜌=−∞. (3.10) Notice that there exist a point 𝜌^∗∈𝒮 and a nonzero vector 𝑥(𝜌^∗)∈𝒳_𝑎. For any given 𝜌, the inequality 𝛼₀𝜌>𝜌^∗ always holds as 𝛼₀>0 becomes large enough. Then for any 𝛼≥𝛼₀, it follows from Lemma 2.1 and 2.2 that 𝛼𝜌∈𝒮 and 𝑥(𝛼𝜌) stays in 𝒳_𝑎, that is, 𝑥²_𝑖(𝛼𝜌)≤ℓ_𝑖forall𝑖. It means that there exists a large positive real 𝐿 such that |𝑃(𝛼𝜌)|≤𝐿 for 𝛼≥𝛼₀ since 𝑃(𝑥) is twice continuously differentiable in ℛ^𝑛.
By Lemma 2.2, we have 𝑑𝑃^𝑑𝛼𝜌=𝑑𝛼∇𝑃^𝑑𝛼𝜌^𝑇𝜌=^𝑛_𝑖=1𝜌_𝑖2𝑥²_𝑖𝛼𝜌−ℓ_𝑖,𝑑²𝑃^𝑑𝛼𝜌𝑑𝛼²=−𝜌^𝑇Diag𝑥𝛼𝜌∇²𝑃𝑥𝛼𝜌+Diag𝛼𝜌⁻¹Diag𝑥𝛼𝜌𝜌, (3.11) where Diag(𝑥(𝛼𝜌))𝜌=(𝜌₁𝑥₁(𝛼𝜌),𝜌₂𝑥₂(𝛼𝜌),…,𝜌_𝑛𝑥_𝑛(𝛼𝜌))^𝑇. For any 𝛼≥𝛼₀, by the definition of 𝒮, it is easy to see that 𝑑²𝑃^𝑑(𝛼𝜌)/𝑑𝛼²≤0, namely, 𝑑𝑃^𝑑(𝛼𝜌)/𝑑𝛼 monotonously decreases on [𝛼₀,+∞). Moreover, since 𝑥(𝛼𝜌)∈𝒳_𝑎, we have 𝑑𝑃^𝑑(𝛼𝜌)/𝑑𝛼≤0 by (3.11). Then, 𝑃^𝑑(𝛼𝜌) is monotonously decreasing on [𝛼₀,+∞). Thus, to prove (3.10), it is only needed to show that there exists a positive real 𝛼≥𝛼₀ such that 𝑑𝑃^𝑑𝛼𝜌𝑑𝛼=^𝑛_𝑖=1𝜌_𝑖2𝑥²_𝑖𝛼𝜌−ℓ_𝑖<0, (3.12) which implies that 𝑃^𝑑(𝛼𝜌) is strictly monotonously decreasing on [𝛼,+∞).
Suppose that ^𝑛_𝑖=1𝜌_𝑖2𝑥²_𝑖𝛼₁𝜌−ℓ_𝑖=0 (3.13) at a point 𝛼₁≥𝛼₀. Since 𝜌_𝑖[𝑥²_𝑖(𝛼₁𝜌)−ℓ_𝑖]≤0forall𝑖 and 𝜌≠0, the equation 𝑥²_𝑟(𝛼₁𝜌)=ℓ_𝑟 holds for some subscript 𝑟, which means that 𝜌_𝑟𝑥_𝑟(𝛼₁𝜌)≠0. By positive definiteness of [∇²𝑃(𝑥(𝛼₁𝜌))+Diag(𝛼₁𝜌)], we have 𝑑𝑃^𝑑(𝛼₁𝜌)/𝑑𝛼₁=0 and 𝑑²𝑃^𝑑(𝛼₁𝜌)/𝑑𝛼²₁<0. One can verify that 𝑑𝑃^𝑑(𝛼𝜌)/𝑑𝛼<0 on (𝛼₁,+∞) since its monotonicity. Obviously, it is very easy to hold (3.12) if there does not exist 𝛼 satisfying (3.13). Consequently, there always exists a positive real 𝛼 such that 𝑃^𝑑(𝛼𝜌) is strictly monotonously decreasing on [𝛼,+∞). It leads to the conclusion (3.10).
Since 𝑃^𝑑∶𝒮→ℛ is concave and the condition (3.10) holds, if (3.9) holds, then the canonical dual function 𝑃^𝑑(𝜌) is coercive on the open convex set 𝒮. Therefore, the canonical dual problem (𝑃^𝑑) has one maximizer 𝜌∈𝒮 by the theory of convex analysis [4, 18]. This completes the proof.

Clearly, when ∇²𝑃(𝑥)>0 on 𝒳_𝑎, the dual feasible space 𝒮 is equivalent to ℛ^𝑛₊={𝜌∈ℛ^𝑛∣𝜌≥0} and Diag(𝜌)∈ℛ^𝑛×𝑛 by (2.1). Notice that lim_𝛼→∞𝑃^𝑑(𝛼𝜌)=−∞ for any given 𝜌≥0. Then 𝑃^𝑑(𝜌) is concave and coercive on ℛ^𝑛₊, and (𝑃^𝑑) has at least one maximizer on ℛ^𝑛₊. In this case, it is then of interest to characterize a unique solution of (𝒫) by the dual variable.

Let 𝐼=𝑖∈{1,2,…,𝑛}∣𝑥²_𝑖(0)<ℓ_𝑖{},𝐽=1,2,…,𝑛𝐼. (3.14)

Theorem 3.4. If ∇²𝑃(𝑥)>0 on 𝒳_𝑎, the primal problem (𝒫) has a unique global minimizer 𝑥(𝜌) determined by 𝜌∈ℛ^𝑛₊ satisfying 𝜌_𝑖=0,∀𝑖∈𝐼,𝑥²_𝑖𝜌=ℓ_𝑖,∀𝑖∈𝐽. (3.15)

Proof. To prove Theorem 3.4, by Theorem 2.3 and 3.1, it is only needed to prove that 𝜌 is a KKT point of (𝑃^𝑑) in ℛ^𝑛₊. By (3.14), the relations 𝑥²_𝑖(𝜌)<ℓ_𝑖 for all 𝑖∈𝐼 also hold. Since 𝜌 satisfies equations 𝑥²_𝑖(𝜌)=ℓ_𝑖 for all 𝑖∈𝐽, we can verify that 𝑥(𝜌) stays in 𝒳_𝑎 and the complementarity conditions 𝜌_𝑖(𝑥²_𝑖(𝜌)−ℓ_𝑖)=0,forall𝑖 hold. Thus, 𝜌 is a KKT point of (𝑃^𝑑) in ℛ^𝑛₊ by (2.9), (2.15), and 𝑥(𝜌) is a unique global minimizer of (𝒫). This completes the proof.

Before beginning of applications to optimal control problems, we present two examples to find global minimizers by differential flows.

Example 3.5. As a particular example of (𝒫), let us consider the following one dimensional nonconvex minimization problem with a box: (𝑥)=1min𝑃3𝑥³+2𝑥s.t.𝑥²≤1. (3.16) We have 𝑃^′(𝑥)=𝑥²+2and𝑃^′′(𝑥)=2𝑥,forall𝑥²≤1. By choosing 𝜌^∗=6√2, we solve the following equation in {𝑥²≤1}: 𝑥²+2+6√2𝑥=0 (3.17) to get a solution 𝑥^∗=−2/(4+3√2). Next we solve the following boundary value problem of the ordinary differential equation: 𝑥̇=−𝑥6√2𝑥+𝜌,𝑥2=−24+3√2,2<𝜌≤6√2. (3.18) To find a parameter such that 𝑥²𝜌=1, (3.19) we get 𝜌=3, (3.20) which satisfies 𝑃^′′(𝑥)+𝜌=2𝑥+3>0,∀𝑥²≤1. (3.21) Let 𝑥(3) be denoted by 𝑥. To find the value of 𝑥, we compute the solution of the following algebra equation: 𝑥²+2+3𝑥=0,𝑥²=1 (3.22) and get 𝑥=−1. It follows from Theorem 3.1 that 𝑥=−1 is a global minimizer of Example 3.5.

Remark 3.6. In this example, we see that a differential flow is useful in solving a nonconvex optimization problem. For the global optimization problem, people usually compute the global minimizer numerically. Even in using canonical duality method, one has to solve a canonical dual problem numerically. Nevertheless, the differential flow directs us to a new way for finding a global minimizer. Particularly, one may expect an exact solution of the problem provided that the corresponding differential equation has an analytic solution.

Example 3.7. Given a symmetric matrix 𝐴∈ℛ^𝑛×𝑛 and a vector 𝑐∈ℛ^𝑛. Let 𝑃(𝑥)=(1/2)𝑥^T𝐴𝑥−𝑐^T𝑥. We consider the following box-constrained nonconvex global optimization: (𝑥)=1min𝑃2𝑥^𝑇𝐴𝑥−𝑐^𝑇𝑥s.t.−ℓ_𝑖≤𝑥_𝑖≤ℓ_𝑖,𝑖=1,2,…,𝑛. (3.23) Since 𝐴 is an indefinite matrix, we choose a large 𝜌^∗∈ℛ^𝑛 such that 𝐴+Diag(𝜌^∗)>0 and sup_{−ℓ𝑖≤𝑥𝑖≤ℓ𝑖}|(𝐴𝑥−𝑐)_𝑖|<𝜌^∗_𝑖ℓ_𝑖,𝑖=1,2,…,𝑛. We see that the differential equation is 𝑑𝑥_𝑖𝜌=𝐴+Diag𝜌_𝑖1𝜌𝑥₁𝜌𝑑𝜌₁+𝐴+Diag𝜌_𝑖2𝜌𝑥₂𝜌𝑑𝜌₂+⋯+𝐴+Diag𝜌_𝑖𝑛𝜌𝑥_𝑛𝜌𝑑𝜌_𝑛,𝑥𝜌^∗=𝐴+Diag𝜌^∗−1𝑐,𝜌∈𝒮∩0≤𝜌≤𝜌^∗, (3.24) where 𝒮={𝜌≥0∣𝐴+Diag(𝜌)>0}. It leads a differential flow 𝑥𝜌=𝐴+Diag𝜌⁻¹𝑐,𝜌∈𝒮. (3.25) For simplicity without generality, we assume that 𝐴=^𝑎1𝑎2_𝑎2𝑎3 and  𝑐=^𝑐1_𝑐1. If for 𝜌, det[𝐴+Diag(𝜌)]≠0, we have 𝐴+Diag𝜌⁻¹=𝑎₃+𝜌₂𝑎₁+𝜌₁𝑎₃+𝜌₂−𝑎²₂−𝑎₂𝑎₁+𝜌₁𝑎₃+𝜌₂−𝑎²₂−𝑎₂𝑎₁+𝜌₁𝑎₃+𝜌₂−𝑎²₂𝑎₁+𝜌₁𝑎₁+𝜌₁𝑎₃+𝜌₂−𝑎²₂. (3.26) Then the dual problem can be formulated as 𝑃𝜌1max=−2𝑐^𝑇𝐴+Diag𝜌⁻¹𝑐−ℓ^𝑇𝜌∶𝜌∈𝒮. (3.27)
If we choose 𝑎₁=−0.5,𝑎₂=−0.5,𝑎₃=−0.3,𝑐₁=𝑐₂=0.3 and ℓ₁=0.5,ℓ₂=2, this dual problem has only one KKT point 𝜌=(1.8,0.7)^𝑇∈𝒮. By Theorem 3.1, 𝑥=[𝐴+Diag(𝜌)]⁻¹𝑐=(1,2)^𝑇 is a global minimizer of Example 3.7 and 𝑃(1,2)=−2.75=𝑃^𝑑(1.8,0.7).

4. Applications to Constrained Optimal Control Problems

In this section, we consider the following constrained linear-quadratic optimal control problem: ()=1min𝐽𝑥,𝑢2^𝑇₀𝑥^𝑇(𝑡)𝑄𝑥(𝑡)+𝑢^T(𝑡)(𝑡)𝑥̇(𝑡)(𝑡)(𝑡)(0)𝑅𝑢𝑑𝑡s.t.=𝐴𝑥+Bu,𝑥=𝑥₀,𝑢(𝑡)∈𝑈,𝑡∈[0,𝑇], (4.1) where 𝑄∈ℛ^𝑛×𝑛,𝑅∈ℛ^𝑚×𝑚 are positive semidefinite and positive definite symmetric matrices respectively, 𝑥(𝑡)∈ℛ^𝑛 is a state vector, and 𝑢(𝑡)∈ℛ^𝑚 is integrable or piecewise continuous on [0,𝑇] within 𝑈. Simply, 𝑈={𝑢∈ℛ^𝑚|−1≤𝑢_𝑖≤1}, and 𝑈 is a unit box. Problems of the above type arise naturally in system science and engineering with wide applications [19, 20].

It is well known that the central result in the optimal control theory is the Pontryagin maximum principle providing necessary conditions for optimality in very general optimal control problems.

4.1. Pontryagin Maximum Principle

Define the Hamilton-Jacobi-Bellman function 𝐻(𝑥,𝑢,𝜆)=𝜆^𝑇()+1𝐴𝑥+𝐵𝑢2𝑥^𝑇1𝑄𝑥+2𝑢^𝑇𝑅𝑢. (4.2) If the control 𝑢 is an optimal solution for the problem (4.1), with 𝑥(⋅) and 𝜆(⋅) denoting the state and costate corresponding to 𝑢(⋅), respectively, then 𝑢 is an extremal control, that is, we have 𝑥̇=𝐻_𝜆𝑥,𝑢,𝜆=𝐴𝑥+𝐵𝑢,𝑥(0)=𝑥₀,̇𝜆=−𝐻_𝑥𝑥,𝑢,𝜆=−𝐴^𝑇𝜆−𝑄𝑥,𝜆(𝑇)=0, (4.3) and a.e.𝑡∈[0,𝑇], 𝐻𝑡,𝑥(𝑡),𝑢(𝑡),𝜆(𝑡)=min_𝑢∈𝑈𝐻𝑡,𝑥(𝑡),𝑢(𝑡),𝜆(𝑡). (4.4)

Unfortunately, above conditions are not, in general, sufficient for optimality. In such a case, we need to go through the process of comparing all the candidates for optimality that the necessary conditions produce, and picking out an optimal solution to the problem. Nevertheless, Lemma 4.1 can prove that the solution satisfies sufficiency conditions of the type considered in this section, then these conditions will ensure the optimality of the solution.

Lemma 4.1. Let 𝑢(⋅) be an admissible control, 𝑥(⋅) and 𝜆(⋅) be the corresponding state and costate. If 𝑥(𝑡), 𝑢(𝑡), and 𝜆(𝑡) satisfy the Pontryagin maximum principle ((4.3)-(4.4)), then 𝑢(𝑡) is an optimal control to the problem (4.1).

Proof. For any given 𝑥,𝜆, let 𝐻^∗(𝑥,𝜆)=min_𝑢∈𝑈𝐻(𝑥,𝑢,𝜆). (4.5) For any 𝑢∈𝑈, by the definition of 𝐻^∗, 𝐻^∗(𝑥,𝜆)≤𝐻(𝑥,𝑢,𝜆), and 𝐻^∗(𝑥,𝜆) is equivalent to the following global optimization min_𝑢∈𝑈12𝑢^𝑇𝑅𝑢+𝜆^𝑇𝐵𝑢. (4.6) Moreover, we can derive an analytic form of the global minimizer for (4.6) via the co-state 𝜆. It is easy to see that the minimizer 𝑢 of (4.6) doesn't depend on 𝑥, that is, 𝜕𝑢/𝜕𝑥=0 which implies that 𝐻^∗_𝑥(𝑥,𝜆)=𝐻_𝑥(𝑥,𝑢,𝜆)+𝐻_𝑢()𝜕𝑢𝑥,𝑢,𝜆𝜕𝑥=𝐻_𝑥(𝑥,𝑢,𝜆). (4.7) Since 𝑈 is a closed convex set, by the classical linear systems theory, the state set 𝑋 of (4.1) is a convex subset of ℛ^𝑛. By the fact that the minimizer 𝑢 does not depend on 𝑥 and the convexity of the integrand in the cost functional, the function 𝐻^∗(𝑥,𝜆) is convex with respect to 𝑥 over 𝑋. In other words, for any 𝑥∈𝑋, and a.e.𝑡∈[0,𝑇], 𝐻^∗𝑥,𝜆(t)≥𝐻^∗𝑥(𝑡),𝜆(𝑡)+𝐻^∗_𝑥^𝑇𝑥(𝑡),𝜆(𝑡)(𝑥(𝑡𝑥(𝑡),𝑢(𝑡),𝜆(𝑡)𝑥−))=𝐻+𝐻^𝑇_𝑥𝑥(𝑡),𝑢(𝑡),𝜆(𝑡)(𝑥−𝑥(𝑡)). (4.8) Thus, for any admissible pair (𝑥(⋅),𝑢(⋅)), and a.e.𝑡∈[0,𝑇], by (4.5), we have 𝐻^∗𝑥(𝑡),𝑢(𝑡),𝜆(𝑡)≥𝐻^∗𝑥(𝑡),𝜆(𝑡)≥𝐻𝑥(𝑡),𝑢(𝑡),𝜆(𝑡)+𝐻^𝑇_𝑥𝑥(𝑡),𝑢(𝑡),𝜆(𝑡)(𝑥(𝑡)−𝑥(𝑡)), (4.9) which leads to 𝐻𝑥(𝑡),𝑢(𝑡),𝜆(𝑡)𝑥(𝑡)(𝑡),𝜆(𝑡)≤̇−𝐻,𝑢𝜆^𝑇(𝑡)(𝑥(𝑡)−𝑥(𝑡)). (4.10) Notice that 𝜆(𝑇)=0 and 𝑥(0)=𝑥(0)=𝑥₀. We can obtain 𝐽[𝑢(⋅)]−𝐽[𝑢(⋅)]=^𝑇₀𝐻𝑥(𝑡),𝑢(𝑡),𝜆(𝑡)𝑥(𝑡)(𝑡),𝜆(𝑡)+−𝐻,𝑢𝑑𝑡_𝑇0𝜆^𝑇(𝑡)𝑥̇(𝑡)−𝑥̇(𝑡)𝑑𝑡≤^𝑇₀𝑑𝜆^𝑇(𝑡𝑥(𝑡)−𝑥(𝑡𝜆)())=−^𝑇(0)(𝑥(0)−𝑥(0))=0. (4.11) This means that 𝐽 attains its minimum at 𝑢. The proof is completed.

Lemma 4.1 reformulates the constrained optimal control problem (4.1) into a global optimization problem (4.6). Based on Theorem 3.4, an analytic solution of (4.1) can be expressed via the co-state.

Theorem 4.2. Suppose that 𝑢=argmin_𝑢∈𝑈12𝑢^T𝑅𝑢+𝜆^T𝐵𝑢. (4.12) We have the following expression 𝑢=−𝑅+Diag𝜌(𝜆)⁻¹𝐵^T𝜆, (4.13) where 𝜌(𝜆) with respect to the co-state 𝜆 is given by 𝜌(𝜆)≥0 satisfying 𝜌_𝑖(𝜆)=0if𝑅⁻¹𝐵^𝑇𝜆²_𝑖𝜌(𝜆)<1,𝑅+Diag⁻¹𝐵^T𝜆²_𝑖=1if𝑅⁻¹𝐵^T𝜆²_𝑖≥1. (4.14)

Proof. The proof of Theorem 4.2 is parallel to Theorem 3.4.

Substituting 𝑢=−[𝑅+Diag(𝜌(𝜆))]⁻¹𝐵^𝑇𝜆 into (4.3), we have 𝑥̇=𝐴𝑥+𝐵−𝑅+Diag𝜌(𝜆)⁻¹𝐵^𝑇𝜆,𝑥(0)=𝑥₀,𝜆̇=−𝐴^𝑇𝜆−𝑄𝑥,𝜆(𝑇)=0. (4.15) If (𝑥(⋅),𝜆(⋅)) is a solution of the above equations (4.15), let 𝑢(⋅)=−𝑅+Diag𝜌𝜆(⋅)⁻¹𝐵^𝑇𝜆(⋅). (4.16) By Lemma 4.1, 𝑥(⋅),𝑢(⋅),𝜆(⋅) satisfy the Pontryagin maximum principle, and we present an analytic form of the optimal control to (4.1) via the canonical dual variable 𝑢(𝑡)=−𝑅+Diag𝜌𝜆(𝑡)⁻¹𝐵^𝑇𝜆(𝑡),a.e.𝑡∈[0,𝑇]. (4.17)

Next, we give an example to illustrate our results.

Example 4.3. We consider (0)=11𝐴=27−2.53,𝐵=2−51.57,𝑄=2.43−2−1,𝑅=2003,𝑥, (4.18) and 𝑇=1 in (4.1). 𝑄≥0 and 𝑅>0 satisfy the assumption in (4.1).
Following the idea of Lemma 4.1 and Theorem 4.2, we need to solve the following boundary value problem for differential equations to derive the optimal solution 𝑥̇=𝐻_𝜆𝜆𝑥(0)=11,̇𝑥,𝑢,=𝐴𝑥+𝐵𝑢,𝜆=−𝐻_𝑥𝑥,𝑢,𝜆=−𝐴^𝑇𝜆(1)𝑢𝜆−𝑄𝑥,=0,₁𝐵=−^𝑇𝜆₁max2,𝐵^𝑇𝜆₁,𝑢₂𝐵=−^𝑇𝜆₂max3,𝐵^𝑇𝜆₂,a.e.𝑡∈[0,1]. (4.19) By solving equations (4.19) in MATLAB, we can obtain the optimal control 𝑢 and the dual variable 𝜌(𝜆) as follows (see Figures 1 and 2).

Figure 1 

The optimal control 𝑢 in Example 4.3.

Figure 2 

The dual variable 𝜌(𝜆) in Example 4.3.

Acknowledgments

The authors would like to thank the referees for their helpful comments on the early version of this paper. This work was supported by the National Natural Science Foundation of China (no. 10971053).