We study a new class of optimization problems called stochastic separated continuous conic programming (SSCCP). SSCCP is an extension to the optimization model called separated continuous conic programming (SCCP) which has applications in robust optimization and sign-constrained linear-quadratic control. Based on the relationship among SSCCP, its dual, and their discretization counterparts, we develop a strong duality theory for the SSCCP. We also suggest a polynomial-time approximation algorithm that solves the SSCCP to any predefined accuracy.

1. Introduction

Stochastic programming is one of the branches of optimization which enjoys a fast development in recent years. It tries to find optimal decisions in problems involving uncertain data, so it is also called “optimization under uncertainty” [1]. Since the problems in reality often involve uncertain data, stochastic programming has a lot of applications.

Many deterministic optimization models have their stochastic counterpart; for example, the stochastic counterpart of linear programming is stochastic linear programming. In this paper, we consider the stochastic counterpart of a kind of optimization model called separated continuous conic programming (SCCP) which has the following form:
(1)(SCCP)max∫0T[(γ+(T-t)c)′u(t)+d′x(t)]dt(SCCP)s.t.α+ta-∫0tGu(s)ds-Fx(t)∈𝒦1,(SCCP)s.t.b-Hu(t)∈𝒦2,(SCCP)s.t.u(t)∈𝒦3,x(t)∈𝒦4,t∈[0,T].
Here the control and state variables (both are decision variables), u(t) and x(t), are vectors of bounded measurable functions of time t∈[0,T]. 𝒦i, i=1,2,3,4, are closed convex cones in the Euclidean space with appropriate dimensions, γ,c,d,α,a,b are vectors, G,F,H are matrices, and the superscript ′ denotes the transpose operation.

SCCP was first studied by Wang et al. [2]. They developed a strong duality theory for SCCP under some mild and verifiable conditions and suggested an approximation algorithm to solve SCCP with predefined precision. SCCP has a variety of applications in robust optimization and sign-constrained linear-quadratic control. However, many applications of SCCP are stochastic in nature in the sense that the values of some parameters in the resulted SCCP models may change over time with some probability distribution. To incorporate this kind of randomness into the SCCP model, we introduce the following stochastic counterpart of SCCP which we call stochastic separated continuous conic programming (SSCCP) problem:
(2)max∫0T1[(γ+(T-t)c)′u(t)+d′x(t)]dt+Eξ(∫T1T2[(γ(ξ)+(T-t)c(ξ))′u(t)+d(ξ)′x(t)]dt)s.t.α+ta-∫0tGu(s)ds-Fx(t)∈𝒦1,t∈[0,T1],b-Hu(t)∈𝒦2,t∈[0,T1],u(t)∈𝒦3,x(t)∈𝒦4,t∈[0,T1],α+T1a+(t-T1)a(ξ)-∫0T1Gu(s)ds-∫T1tGu(s)ds-Fx(t)∈𝒦(ξ)1,-∫T1tGu(s)ds-Fxt∈[T1,T2],b(ξ)-Hu(t)∈𝒦(ξ)2,t∈[T1,T2],u(t)∈𝒦(ξ)3,x(t)∈𝒦(ξ)4,t∈[T1,T2],
where ξ is a random variable.

SSCCP is formulated with the similar idea as that of the stochastic linear programming [1, 3]. There are two stages in this problem; the values of some parameters in the second stage depend on the value of a random variable ξ.

Our goal in this paper is developing the strong duality for SSCCP and suggesting a solution method to solve it approximately with predefined precision. Here is a summary of our main results. Through discretization, we connect SSCCP and its dual to two ordinary conic programs, and we show that strong duality holds for SSCCP and its dual under some mild (and verifiable) conditions on these two ordinary conic programs. Furthermore, the optimal values of those two conic programs provide an explicit bound on the duality gap between SSCCP and its dual, based on which we suggest a polynomial-time approximation algorithm that solves SSCCP to any predefined accuracy. According to our knowledge, we are the first to raise the SSCCP model and there have been no other results on SSCCP besides those in this paper.

The paper is organized as follows. In Section 2, we present an overview on the related literature. We also give a concrete example to show the application of SSCCP. In Section 3, we construct a dual for SSCCP. We also discretize SSCCP and its dual into two ordinary conic programs, and bring out their relations. In Section 4, we discuss the strong feasibility for SSCCP, its dual, and their discretizations. We then establish the strong duality result for SSCCP and its dual in Section 5. This leads to a polynomial-time approximation algorithm with an explicit error bound, detailed in Section 6. In Section 7, we summarize what we get for SSCCP and point out some future research directions.

For simpler presentation, in the remainder of this paper, we will concentrate on the following problem, which is the corresponding SSCCP when ξ is a discrete variable and only takes two different values with probability θ and 1-θ, that is, there are only two scenarios in the second stage of SSCCP:
(3)max∫0T1[(γ+(T-t)c)′u(t)+d′x(t)]dt+∫T1T[(γ˘1+(T-t)c˘1)′v(t)+d˘1′y(t)]dt+∫T1T[(γ˘2+(T-t)c˘2)′w(t)+d˘2′z(t)]dts.t.α+ta-∫0tGu(s)ds-Fx(t)∈𝒦1,t∈[0,T1],b-Hu(t)∈𝒦2,t∈[0,T1],u(t)∈𝒦3,x(t)∈𝒦4,t∈[0,T1],α1+ta1-∫0T1Gu(s)ds-∫T1tGv(s)ds-Fy(t)∈𝒦11,t∈(T1,T],b1-Hv(t)∈𝒦21,t∈(T1,T],v(t)∈𝒦31,y(t)∈𝒦41,t∈(T1,T],α2+ta2-∫0T1Gu(s)ds-∫T1tGw(s)ds-Fz(t)∈𝒦12,-∫T1tGw(s)ds-Ft∈(T1,T],b2-Hw(t)∈𝒦22,t∈(T1,T],w(t)∈𝒦32,z(t)∈𝒦42,t∈(T1,T],
where the first-stage control and state variables are u(t) and x(t), t∈[0,T1], and the second-stage control and state variables are v(t),w(t), y(t), and z(t), t∈(T1,T]. Also α1=α+T1a-T1a1, α2=α+T1a-T1a2, γ˘1=θγ1, c˘1=θc1, d˘1=θd1, γ˘2=(1-θ)γ2, c˘2=(1-θ)c2, d˘2=(1-θ)d2.

Note that although (3) is a deterministic optimization problem, it is not an SCCP. To see why this is the case, one can try to formulate (3) into the form of SCCP and it then becomes clear that (3) cannot fit into the SCCP form.

In the rest of this paper, we will use some results on conic programming without explanations. Interested readers can consult the books on conic programming (e.g., [4]) for the related results.

2. Literature Review

Bellman [5, 6] first introduced the so-called continuous linear programming (CLP), which has the following form:
(4)(CLP)max∫0Tc(t)′x(t)dt(CLPL)s.t.Bx(t)-∫0tKx(s)ds≤b(t)(CLPL)s.t.x(t)≥0,t∈[0,T].
Here x(t) is a decision variable. The model has wide-ranging applications (e.g., the bottleneck problem [5]). But CLP is very difficult to solve in its general form. Later, Anderson [7] introduced separated continuous linear programming (SCLP) (see (5)), a special case of CLP, to model the job-shop scheduling problems:
(5)(SCLP)max∫0T[(γ+(T-t)c)′u(t)+d′x(t)]dt(SCLPL)s.t.∫0tGu(s)ds+Fx(t)≤α+ta,(SCLPL)s.t.Hu(t)≤b,(SCLPL)s.t.u(t)≥0,x(t)≥0,t∈[0,T].
The word “separated” refers to the fact that there are two kinds of constraints in SCLP: the constraints involving integration and the instantaneous constraints [7].

Anderson et al. [8] studied the properties of the extreme solutions of the SCLP, based on which Anderson and Philpott [9] developed a simplex type of algorithm for a network-based SCLP. Refer to Anderson and Philpott [10] and Anderson and Nash [11] for their other results on SCLP. Pullan [12–18] continues studying SCLP in a series of papers. He systematically developed a duality theory and solution algorithms for the SCLP.

There are other researches focused on other forms of SCLP, including Luo and Bertsimas [19], Shapiro [20], Fleischer and Sethuraman [21], Weiss [22], and Nasrabadi et al. [23].

One of the extensions of SCLP is SCCP introduced by Wang et al. [2] in which the constraints involve the convex cone in their right hand side. When all the convex cones are nonnegative orthants, SCCP reduces to SCLP. In [2], based on the relationship among SCCP, its dual, and their discretization counterparts, they develop a strong duality theory for the SCCP. They also suggest a polynomial-time approximation algorithm that solves the SCCP to any predefined accuracy.

Wang [24, 25] extends SCCP to generalized separated continuous conic programming (GSCCP) by allowing the parameters in (1) to be piece-wise constants and extends the results of [2] for SCCP to GSCCP. In this paper, we extend SCCP to SSCCP by allowing the changes of values of some parameters in SCCP in the second stage. We also extend the results of [2] for SCCP to SSCCP.

2.1. A Motived Example for SSCCP

We consider a problem which appears in [2]; for completeness, we reproduce the problem description and the formulation below.

A network processes a continuous flow of jobs at two machines. The jobs visit machines 1 and 2 in the order 1→2→1, that is, a total of three processing steps; see Figure 1. Corresponding to each processing step, there is a buffer holding the fluid. At t=0, the initial levels of fluid at the three steps are 50, 20, and 120 units. The input rates of fluid from outside to the three buffers are 0.01, 0.01, and 0.01. To process each unit of job (“fluid”), the time requirements at the three steps are 0.4, 0.8, and 0.2 time units.

The fluid network in the example in Section 2.1.

The problem is to find the processing rates at the three steps, ui(t), i=1,2,3, which determine the fluid levels in the three buffers, xi(t), i=1,2,3, during a given time interval [0,T] such that the fluid levels in the three buffers are maintained as close as possible to a prespecified constant level d=(301080)′.

The problem can be formulated as follows:
(6)min∫0T[(x(t)-d)′(x(t)-d)]dts.t.∫0tGu(s)ds+x(t)=α+ta,b-Hu(t)≥0,u(t)≥0,x(t)≥0,t∈[0,T],
where
(7)G=(100-1100-11),H=(0.400.200.80),α=(5020120),a=(0.010.010.01),b=(11).
We can further express the above problem in the form of SCCP. Please refer to [2] for the details.

In reality, the values of a and b could be changed during [0,T] for example, when the machine 1 experiences partial breakdown within [T1,T]; where 0≤T1≤T, the corresponding value of capacity vector for machine 1, b1, will change during [T1,T]. This makes the formulation of the problem an SSCCP. We omit the details here.

3. The Dual and Discretizations3.1. The Dual

The dual of SSCCP that we will focus on is the following problem:
(8)(SSCCP*)min∫0T-T1[(α1+(T-t)a1)′h(t)+b1′l(t)]dt+∫0T-T1[(α2+(T-t)a2)′p(t)+b2′q(t)]dt+∫T-T1T[(α+(T-t)a)′f(t)+b′g(t)]dts.t.∫0tG′h(s)ds+H′l(t)-(γ˘1+tc˘1)∈𝒦31*,s.t.∫0tG′h(s)ds+H′l(t)-t∈[0,T-T1],F′h(t)-d˘1∈𝒦41*,t∈[0,T-T1],h(t)∈𝒦11*,l(t)∈𝒦21*,t∈[0,T-T1],∫0tG′p(s)ds+H′q(t)-(γ˘2+tc˘2)∈𝒦32*,s.t.∫0tG′h(s)ds+H′l(t)-t∈[0,T-T1],F′p(t)-d˘2∈𝒦42*,t∈[0,T-T1],p(t)∈𝒦12*,q(t)∈𝒦22*,t∈[0,T-T1],∫0T-T1G′(h(s)+p(s))ds+∫T-T1tG′f(s)ds+H′g(t)-(γ+tc)∈𝒦3*,s.t.∫0tG′h(s)ds+H′l(t)-t∈(T-T1,T],F′f(t)-d∈𝒦4*,t∈(T-T1,T],f(t)∈𝒦1*,g(t)∈𝒦2*,t∈(T-T1,T],
where the decision variables h(t), l(t), p(t), q(t), f(t), and g(t) are bounded measurable functions. 𝒦i*, 𝒦ij* are the dual cones of 𝒦i and 𝒦ij, i=1,2,3,4, j=1,2, respectively.

The derivation of the above dual problem is similar to the derivation of the dual problem for LP (see, e.g., [26]) and we omit the details here. Because SSCCP involves time, to achieve some degree of symmetry in the dual (to facilitate the later analysis), we choose to write the dual in the reversed time; that is, t in the dual is T-t in the primal.

The following weak duality is readily shown from the derivation of (SSCCP*).

Proposition 1.

The weak duality holds between SSCCP and SSCCP*; that is, if u(t),x(t), t∈[0,T1], v(t), y(t), w(t), z(t), t∈(T1,T] is a feasible solution for SSCCP and h(t), l(t), p(t), q(t), t∈[0,T-T1), f(t), g(t), t∈[T-T1] is a feasible solution for SSCCP*, then
(9)∫0T1[(γ+(T-t)c)′u(t)+d′x(t)]dt+∫T1T[(γ˘1+(T-t)c˘1)′v(t)+d˘1′y(t)]dt+∫T1T[(γ˘2+(T-t)c˘2)′w(t)+d˘2′z(t)]dt≤∫0T-T1[(α1+(T-t)a1)′h(t)+b1′l(t)]dt+∫0T-T1[(α2+(T-t)a2)′p(t)+b2′q(t)]dt+∫T-T1T[(α+(T-t)a)′f(t)+b′g(t)]dt.

Next we will introduce the discretizations for SSCCP and SSCCP*, respectively, and discuss the relationships among SSCCP, SSCCP*, and their discretizations. But first, we need the following notation and conventions which mostly follow what is used in [2].

Notation and Conventions

When we say (u(t),x(t),v(t),y(t),w(t),z(t)) is a feasible solution to SSCCP, we mean (u(t),x(t),t∈[0,T1],v(t),y(t),w(t),z(t),t∈(T1,T]) is a feasible solution to SSCCP.

By default, all vectors are column vectors. One exception is when we denote the solutions to SSCCP and its dual (or their variations) as (u,x,v,y,w,z) and (h,l,p,q,f,g), we mean (u′,x′,v′,y′,w′,z′)′ and (h′,l′,p′,q′,f′,g′)′.

π={t0,…,tm1,…,tm1+m2} denotes a partition of [0,T1,T] into m1+m2 segments:
(10)0=t0<t1<⋯<tm1=T1<tm1+1<⋯<tm1+m2=T,
where m1 and m2 are positive integer numbers.

Given a partition π={t0,…,tm} and a vector r^:=(r^(t0),r^(t1),…,r^(tm)), where r^(·) is a right continuous function, the following (continuous) function
(11)r(t)=(ti-tti-ti-1)r^(ti-1)+(t-ti-1ti-ti-1)r^(ti),fort∈[ti-1,ti],i=1,…,m,
is called a piecewise linear extension of r^, whereas the following (right-continuous) function
(12)r(t)={r^(ti-1),t∈[ti-1,ti),fori=1,…,m,r^(tm-1),t=T,
is called a piecewise constant extension of r^.

When (u(t),x(t),v(t),y(t),w(t),z(t)) is a feasible solution to SSCCP, with u(t), v(t), w(t) being piecewise constant and x(t), y(t), z(t) piecewise linear, we assume u(t), v(t), w(t) is right continuous, and x(t), y(t), z(t) is continuous, with y(T1+)=x(T1), z(T1+)=x(T1), and the pieces of both u and x correspond to a common partition for [0,T1], and the pieces of both v,w and y,z correspond to a common partition for (T1,T].

When (h(t),l(t),p(t),q(t),f(t),g(t)) is a feasible solution to SSCCP*, with h(t), p(t), f(t) being piecewise constant and l(t), q(t), g(t) piecewise linear, we assume h(t), p(t), and f(t) are right continuous and l(t), q(t) and g(t) are continuous, with g(T-T1+)=l(T-T1)+q(T-T1), and the pieces of h, l, p, and q correspond to a common partition for [0,T-T1], and the pieces of both f and g correspond to a common partition for (T-T1,T].

For i=1,2,3,4, denote 𝒦i,m:=𝒦i×⋯×𝒦i︸m, and similarly denote 𝒦i,m*:=𝒦i*×⋯×𝒦i*︸m.

3.2. The Discretizations

We start with introducing the following discretization of SSCCP based on the partition π of [0,T1,T], where 0=t0<t1<⋯<tm1=T1<tm1+1<⋯<tm1+m2=T:
(13)(SCP1(π))max∑i=1m1((γ+(T-ti+ti-12)c)′u^i+d′[x^i+x^i-1]ti-ti-12(γ+(T-ti+ti-12)c)′u^i)+∑i=1m2((γ˘1+(T-tm1+i+tm1+i-12)c˘1)′v^i+d˘1′[y^i+y^i-1]tm1+i-tm1+i-12(γ˘1+(T-tm1+i+tm1+i-12)c˘1)′v^i)+∑i=1m2((γ˘2+(T-tm1+i+tm1+i-12)c˘2)′w^i+d˘2′[z^i+z^i-1]tm1+i-tm1+i-12(γ˘1+(T-tm1+i+tm1+i-12)c˘1)′v^i)s.t.α+tia-[Gu^1+⋯+Gu^i+Fx^i]∈𝒦1,s.t.α+tia-s.t.i=1,2,…,m1;(ti-ti-1)b-Hu^i∈𝒦2,i=1,…,m1;u^i∈𝒦3,x^i∈𝒦4,i=1,…,m1;α1+tm1+ia1-[Gu^1+⋯+Gu^m1]-[Gv^1+⋯+Gv^i]-Fy^i∈𝒦11,-[Gv^1+⋯+G]i=1,2,…,m2;(tm1+i-tm1+i-1)b1-Hv^i∈𝒦21,(tm1+i-tm1+i-1)b1i=1,…,m2;v^i∈𝒦31,y^i∈𝒦41,i=1,…,m2;α2+tm1+ia2-[Gu^1+⋯+Gu^m1]-[Gw^1+⋯+Gw^i]-Fz^i∈𝒦12,i=1,2,…,m2;(tm1+i-tm1+i-1)b2-Hw^i∈𝒦22,(tm1+i-tm1+i-1)b2i=1,…,m2;w^i∈𝒦32,z^i∈𝒦42,i=1,…,m2.
Note that here we require that y^0=z^0=x^m1 and
(14)α-Fx^0∈𝒦1,x^0∈𝒦4.
Clearly, (SCP1(π)) is a conic program.

Lemma 2.

From a feasible solution for SCP1(π), one can get a feasible solution for SSCCP with the same objective values, if 𝒦1⊆𝒦11, 𝒦1⊆𝒦12, 𝒦4⊆𝒦41, 𝒦4⊆𝒦42.

Proof.

Suppose (u^,x^,v^,y^,w^,z^) is a feasible solution for SCP1(π). Let
(15)u(t)={u^iti-ti-1,t∈[ti-1,ti),i=1,…,m1,u^m1tm1-tm1-1,t=tm1,x(0)=x^0,x(t)=ti-tti-ti-1x^i-1+t-ti-1ti-ti-1x^i,i=1,…,m1,v(t)=v^itm1+i-tm1+i-1,t∈(tm1+i-1,tm1+i],v(t)=v^itm1+i-tm1+i-1,t∈(Li=1,…,m2,y(T1+)=x^m1,y(t)=tm1+i-ttm1+i-tm1+i-1y^i-1+t-tm1+i-1tm1+i-tm1+i-1y^i,v(t)=v^itm1+i-tm1+i-1,t∈(Li=1,…,m2,w(t)=w^itm1+i-tm1+i-1,t∈(tm1+i-1,tm1+i],v(t)=v^itm1+i-tm1+i-1,t∈(Li=1,…,m2,z(T1+)=x^m1,z(t)=tm1+i-ttm1+i-tm1+i-1z^(ti-1)+t-tm1+i-1tm1+i-tm1+i-1z^i,v(t)=v^itm1+i-tm1+i-1,t∈(LLLi=1,…,m2;
then we have u(t)∈𝒦3, x(t)∈𝒦4, t∈[0,T1], v(t)∈𝒦31, t∈(T1,T], w(t)∈𝒦32, and t∈(T1,T].

Because x^m1∈𝒦4 and y^0=z^0=x^m1, y^0∈𝒦4 and z^0∈𝒦4. When 𝒦4⊆𝒦41, 𝒦4⊆𝒦42, we have y(t)∈𝒦41, and z(t)∈𝒦42, and t∈(T1,T].

For t=0,
(16)α+ta-∫0tGu(s)ds-Fx(t)=α-Fx^0∈𝒦1.

For t∈(t0,t1),
(17)α+ta-∫0tGu(s)ds-Fx(t)=α+ta-Gu^1t1t-F(t1-tt1x^0+tt1x^1)=tt1(α+t1a-Gu^1-Fx^1)+t1-tt1(α-Fx^0)∈𝒦1.

For t∈[ti-1,ti), i=2,…,m1,
(18)α+ta-∫0tGu(s)ds-Fx(t)=α+ta-(∫0t1Gu(s)ds+⋯+∫ti-1tGu(s)ds)-Fx(t)=α+ta-(Gu^1t1-t0(t1-t0)+⋯+Gu^iti-ti-1(t-ti-1))-F(ti-tti-ti-1x^i-1+t-ti-1ti-ti-1x^i)=t-ti-1ti-ti-1(α+tia-(Gu^1+⋯+Gu^i-1+Gu^i)-Fx^i)+ti-tti-ti-1(α+ti-1a-(Gu^1+⋯+Gu^i-1)-Fx^i-1)∈𝒦1.

For t=tm1,
(19)α+ta-∫0tGu(s)ds-Fx(t)=α+tm1a-(Gu^1+⋯+Gu^m1-1+Gu^m1)-Fx^m1∈𝒦1.

For t∈[ti-1,ti), i=1,…,m1,
(20)b-Hu(t)=b-Hu^iti-ti-1=1ti-ti-1((ti-ti-1)b-Hu^i)∈𝒦2.

For t=tm1,
(21)b-Hu(t)=b-Hu^m1tm1-tm1-1=1tm1-tm1-1((tm1-tm1-1)b-Hu^m1)∈𝒦2.

For t∈(tm1,tm1+1],
(22)α1+ta1-∫0T1Gu(s)ds-∫T1tGv(s)ds-Fy(t)=α1+ta1-(Gu^1+⋯+Gu^m1-1+Gu^m1)-Gv^1tm1+1-tm1(t-tm1)-F(tm1+1-ttm1+1-tm1y^0+t-tm1tm1+1-tm1y^1)=t-tm1tm1+1-tm1(α1+tm1+1a1-(Gu^1+⋯+Gu^m1-1=t-tm1tm1+1-tm1α1+tm1+1a1-+Gu^m1)-Gv^1-Fy^1(Gu^1+⋯+Gu^m1-1)+tm1+1-ttm1+1-tm1(α+T1a-(Gu^1+⋯+Gu^m1-1=t-tm1tm1+1-tm1α1+tm1+1a1+Gu^m1)-Fx^m1(Gu^1+⋯+Gu^m1-1)∈𝒦11,when𝒦1⊆𝒦11.

For t∈(tm1+i-1,tm1+i], i=2,…,m2,
(23)α1+ta1-∫0T1Gu(s)ds-∫T1tGv(s)ds-Fy(t)=α1+ta1-(Gu^1+⋯+Gu^m1-1+Gu^m1)-(Gv^1tm1+1-tm1(tm1+1-tm1)+⋯+Gv^itm1+i-tm1+i-1(t-tm1+i-1))-Fy(t)=α1+ta1-(Gu^1+⋯+Gu^m1-1+Gu^m1)-(Gv^1+⋯+Gv^i-1+Gv^itm1+i-tm1+i-1(t-tm1+i-1))-F(tm1+i-ttm1+i-tm1+i-1y^i-1+t-tm1+i-1tm1+i-tm1+i-1y^i)=t-tm1+i-1tm1+i-tm1+i-1×(α1+tm1+ia1-(Gu^1+⋯+Gu^m1-1+Gu^m1)-(Gv^1+⋯+Gv^i-1+Gv^i)-Fy^i(Gu^1+⋯+Gu^m1-1+Gu^m1))+tm1+i-ttm1+i-tm1+i-1×(α1+tm1+i-1a1-(Gu^1+⋯+Gu^m1-1+Gu^m1)-(Gv^1+⋯+Gv^i-1)-Fy^i-1(Gu^1+⋯+Gu^m1-1+Gu^m1))∈𝒦11.
For t∈(tm1+i-1,tm1+i], i=1,…,m2(24)b1-Hv(t)=b1-Hv^itm1+i-tm1+i-1=1tm1+i-tm1+i-1((tm1+i-tm1+i-1)b1-Hv^i)∈𝒦21.
Similarly, if 𝒦1⊆𝒦12, then
(25)α2+ta2-∫0T1Gu(s)ds-∫T1tGw(s)ds-Fz(t)∈𝒦12,α2+ta2-∫0T1Gu(s)ds-∫T1tGww(s)ds-t∈(T1,T],b2-Hw(t)∈𝒦22,t∈(T1,T].
So (u(t),x(t),v(t),y(t),w(t),z(t)) is a feasible solution for SSCCP when 𝒦1⊆𝒦11, 𝒦1⊆𝒦12, 𝒦4⊆𝒦41, and 𝒦4⊆𝒦42.

It is easy to see that the objective value of (u(t),x(t),v(t),y(t),w(t),z(t)) is the same as that of (u^,x^,v^,y^,w^,z^). We omit the details here.

We now introduce the following discretization of SSCCP* based on the partition π′ of [0,T-T1,T], π′={t0,t1,…,tm