On intersection of two mixing sets with applications to joint chance-constrained programs

Abstract

We study the polyhedral structure of a generalization of a mixing set described by the intersection of two mixing sets with two shared continuous variables, where one continuous variable has a positive coefficient in one mixing set, and a negative coefficient in the other. Our developments are motivated from a key substructure of linear joint chance-constrained programs (CCPs) with random right hand sides from a finite probability space. The CCPs of interest immediately admit a mixed-integer programming reformulation. Nevertheless, such standard reformulations are difficult to solve at large-scale due to the weakness of their linear programming relaxations. In this paper, we initiate a systemic polyhedral study of such joint CCPs by explicitly analyzing the system obtained from simultaneously considering two linear constraints inside the chance constraint. We carry out our study on this particular intersection of two mixing sets under a nonnegativity assumption on data. Mixing inequalities are immediately applicable to our set, yet they are not sufficient. Therefore, we propose a new class of valid inequalities in addition to the mixing inequalities, and establish conditions under which these inequalities are facet defining. Moreover, under certain additional assumptions, we prove that these new valid inequalities along with the classical mixing inequalities are sufficient in terms of providing the closed convex hull description of our set. We also show that linear optimization over our set is polynomial-time, and we independently give a (high-order) polynomial-time separation algorithm for the new inequalities. We complement our theoretical results with a computational study on the strength of the proposed inequalities. Our preliminary computational experiments with a fast heuristic separation approach demonstrate that our proposed inequalities are practically effective as well.

Appendices

Appendix A. Proof of Proposition 3.4

Proof

We first establish the necessity of the condition $w_{s_1} = w_{\alpha _1}$ for inequality (9) to be a facet. Suppose $w_{s_1} < w_{\alpha _1}$. Note that inequality (9) given for $S'=\{\alpha _1\rightarrow s_1\rightarrow s_2\rightarrow \dots \rightarrow s_ {\eta }\}$ is simply

$$\begin{aligned} y_p + y_d + \sum _{j = 1}^{\eta } (w_{s_j} - w_{s_{j + 1}} ) z_{s_j}\ge w_{s_1} + (w_{\alpha _1} - w_{s_1})(1 - z_{\alpha _1}). \end{aligned}$$

This inequality is stronger than the original inequality (9) given for $S=\{s_1\rightarrow s_2\rightarrow \dots \rightarrow s_ {\eta }\}$ because $(w_{\alpha _1} - w_{s_1})(1 - z_{\alpha _1})\ge 0$. Hence, this establishes the necessity of condition $w_{s_1} = w_{\alpha _1}$. The argument for the necessity of condition $v_{t_1} = v_{\beta _1}$ for inequality (10) to be a facet is identical.

To see that inequality (9) is facet defining if $w_{s_1} = w_{\alpha _1}$, first, for all $j \in \varOmega {\setminus } S$, we consider the points $(w_{\alpha _1}, 0, \mathbf {e}_{j})$. These points are feasible (see the proof of Proposition 3.3). In addition, these points satisfy inequality (9) at equality and are affinely independent. Next, for all $j \in [\eta ]$, we consider the points $A\Big (\cup _{i = j}^\eta s_i \Big )=(w_{s_{j }},0, \sum _{i \in \varOmega {\setminus } (\cup _{i = j}^\eta s_i)} \mathbf {e}_{s_i})$. The feasibility of these points follow from Lemma 3.1. In addition, these points satisfy inequality (9) at equality and are affinely independent. Finally, we consider the feasible points $A(\emptyset )$ and $C(\varOmega )$, which are affinely independent from all other points. In addition, $A(\emptyset )$ and $C(\varOmega )$ satisfy inequality (9) at equality. Hence, we obtain $m + 2$ affinely independent points that are feasible and satisfy inequality (9) at equality. This proves that inequality (9) is facet-defining for ${{{\mathrm{{conv}}}}}(\mathcal {P})$.

The proof for inequality (10) to be facet defining when $v_{t_1} = v_{\beta _1}$ is similar. In this case, we consider the points D, $C(\varOmega )$, $C(\varOmega {\setminus } \{j\})$, for all $j \in \varOmega {\setminus } T$, and $B\Big ( \cup _{i = j}^\rho t_i \Big )$, for all $j \in [\rho ]$ These points are feasible from Lemma 3.1 and are also affinely independent. $\square$

Appendix B. Proof of Proposition 3.5

Proof

If $w_{r_1} < w_{\alpha _{1}}$, then we can attach $\alpha _{1}$ at the beginning of the sequence $\Pi$ to obtain another valid inequality of form (11) (or equivalently (12))

$$\begin{aligned} 2y_p&+ \sum _{j = 1}^{\tau _R} (w_{r_j} -w_{ r_{j + 1}})z_{ r_j}\\&+ \sum _{j = 1}^{\tau _G} (v_{g_j} - v_{g_{j + 1}})z_{g_j} \ge w_{ r_1} + v_{g_1} + (w_{\alpha _{1}} - w_{ r_1})(1 - z_{\alpha _{1}} ). \end{aligned}$$

The resulting inequality is at least as strong as the original inequality because $w_{\alpha _{1}} > w_{ r_1}$ and $1 - z_{\alpha _{1}} \ge 0$. Similarly, if $v_{g_1} < v_{\beta _1}$, then we can attach $\beta _1$ at the beginning of the sequence $\Pi$ to obtain another inequality that is at least as strong as the original inequality. This shows the necessity of the facet conditions.

To see the sufficiency, first consider the feasible points $C(\emptyset )$ and D (see Lemma 3.1 for their feasibility). These points satisfy inequality (11) at equality. Next, we consider the feasible point $C(\varOmega )$, which satisfies inequality (11) at equality. Now, consider the points $(\frac{w_{\alpha _1} + v_{\beta _1}}{2}, \frac{w_{\alpha _1} - v_{\beta _1}}{2}, \mathbf {e}_j )$, for all $j \in \varOmega {\setminus } \Pi$. For each $j \in \varOmega {\setminus } \Pi$, using Observation 3.1(i) and the feasibility of the point $C(\varOmega )=(\frac{w_{\alpha _1} + v_{\beta _1}}{2}, \frac{w_{\alpha _1} - v_{\beta _1}}{2}, \mathbf 0 )$, we conclude that these points are also feasible. Since $j \not \in \Pi$, these points satisfy (11) at equality as well. Note that the points considered thus far are affinely independent.

Next, for all $j \in [\tau ] {\setminus } \{1\}$ such that $\pi _j \in \Pi$, if $w_{\pi _j} < {\bar{w}}_{\Pi ,j}$ and $v_{\pi _j} < {\bar{v}}_{\Pi ,j}$, then we consider the point $(\frac{w_{\alpha _1} + v_{\beta _1}}{2}, \frac{w_{\alpha _1} - v_{\beta _1}}{2}, \mathbf {e}_{\pi _j} )$. For each such j, the feasibility of the associated point follows from the feasibility of $C(\varOmega )$ and Observation 3.1(i). In addition, this point also satisfies inequality (11) at equality, because $(w_{\pi _{j}} - {\bar{w}}_{\Pi ,{j}} )_+ = (v_{\pi _{j}} - {\bar{v}}_{\Pi ,{j}} )_+ = 0$, so the left-hand side of inequality (11), after substituting this point, becomes $w_{\alpha _1} + v_{\beta _1}$. Otherwise, if $w_{\pi _j} \ge {\bar{w}}_{\Pi ,j}$ or $v_{\pi _j} \ge { \bar{v}}_{\Pi ,j}$ for some $j \in [\tau ] {\setminus } \{1\}$, then we consider the following feasible point $C \Big ( \Pi {\setminus } ( \cup _{i = 1}^{j - 1} \{\pi _i\} ) \Big ) = (\frac{{\bar{w}}_{\Pi ,{j - 1}} + {\bar{v}}_{\Pi ,{j - 1}}}{2}, \frac{{\bar{w}}_{\Pi ,{j - 1}} - {\bar{v}}_{\Pi ,{j - 1}}}{2}, \sum _{i = 1}^{j - 1}\mathbf {e}_{\pi _i} + \sum _{i \in ( \varOmega {\setminus } \Pi )} \mathbf {e}_i )$. Note also

$$ \sum _{i = 1}^{j - 1} \Bigg ( \Big (w_{\pi _{i}} - {\bar{w}}_{\Pi ,{i}} \Big )_+ \Bigg ) + {\bar{w}}_{\Pi ,{j - 1}} = \max _{\ell \in [\tau ]}\; w_{\pi _\ell } = w_{\alpha _1}, $$

and

$$ \sum _{i = 1}^{j - 1} \Bigg ( \Big (v_{\pi _{i}} - {\bar{v}}_{\Pi ,i} \Big )_+ \Bigg ) + {\bar{v}}_{\Pi ,{j - 1}}= \max _{\ell \in [\tau ]}\; v_{\pi _\ell } = v_{\beta _1}, $$

because $\alpha _1\in \Pi$ and $\beta _1\in \Pi$. Thus, the point $C \Big ( \Pi {\setminus } ( \cup _{i = 1}^{j - 1} \{\pi _i\} ) \Big )$ satisfies inequality (11) at equality as well. Also, these points are affinely independent from the points listed earlier. Hence, in total, we obtain $m + 2$ affinely independent feasible points that satisfy inequality (11) at equality. This completes the proof. $\square$

Appendix C. Proof of Lemma 3.7

Proof

To prove our claim, first, observe that if $D\in {\widehat{{\mathcal {O}}}}$, then substituting the point D into inequality (10) defined by the sequence T as defined in the premise of the lemma, the left-hand side becomes $-u_d+\sum _{j = 1}^{\rho } (v_{t_j} - v_{t_{j + 1}} ) = v_{t_1}=v_{\beta _1}$ (recall $v_{t_{\rho + 1}}$ in (10)), which proves that inequality (10) defined by T is tight at $D \in {\widehat{{\mathcal {O}}}}$.

Next consider any solution ${\hat{o}}_{i}=B(V_i) \in {\widehat{{\mathcal {O}}}}$ or ${\hat{o}}_{i}=C(V_i) \in {\widehat{{\mathcal {O}}}}$ with $V_i\ne \emptyset$ and $i\in [{\hat{p}}]$. From the definitions of $B(V_i)$ and $C(V_i)$, we have $y_p^i-y_d^i=v_{\bar{j}_i}$ (recall the definition of $\bar{j}_i$) and $z_k^i=1$ for all $k\in \varOmega$ such that $v_k>v_{\bar{j}_i}$ (from inequality (1b) in the original constraint set). Also, by definition of T, $\bar{j}_i=t_{k_i}$ for some $k_i\in [\rho ]$, and we have $v_{t_j}\ge v_{t_{k_i}}$ for $j\in [k_i-1]$; hence, $z^i_{t_j}=1$ for all $j\in [k_i-1]$ such that $v_{t_j}> v_{t_{k_i}}$. Then $\sum _{j=1}^{ k_i} ( v_{t_j} - v_{t_{j + 1}} )z^i_{t_j} = v_{\beta _1} - v_{t_{k_i}}$ where the equality holds because $z_{t_{k_i}}^i=0$, $t_1=\beta _1$ and for $j\in [k_i-1]$ we have $z_{t_j}^i=1$ if $v_{t_j}> v_{t_{k_i}}$. Substituting this term and the relation $y_p^i - y_d^i=v_{\bar{j}_i}=v_{t_{k_i}}$ in inequality (10) leads to the equivalent inequality given by

$$\begin{aligned} v_{t_{k_i}} + v_{\beta _1 } - v_{t_{k_i}} + \sum _{j= k_i+1} ^ {\rho }( v_{t_j} - v_{t_{j + 1}} )z^i_{t_j} \ge v_{\beta _1}. \end{aligned}$$

(27)

Suppose, for contradiction, that ${\hat{o}}_i$ does not satisfy inequality (10) at equality for this choice of T. Then, from (27), we see that we must have $\sum _{j= k_i+1} ^ {\rho }( v_{t_j} - v_{t_{j + 1}} )z^i_{t_j}>0$. In other words, there exists $t_{j'} \in T$ for some $j'\in [\rho ]{\setminus } [k_i]$ with both $z_{t_{j'}}^i=1$ (i.e., $t_{j'}\not \in V_i$) and $v_{t_{j'}} - v_{t_{j' + 1}}>0$. This along with Assumption A3, implies that $v_{t_{j'}}>0$. Moreover, from $j'\in [\rho ]{\setminus }[k_i]$, $t_{k_i}=\bar{j}_i$ and the definition of the sequence T, we deduce $v_{ t_j'} \le v_{t_{k_i}} = v_{\bar{j}_i}$.

Because $t_{j'} \in T{\setminus } V_i$, there exists another point, say ${\hat{o}}_{\ell }=B(V_\ell ) \in {\widehat{{\mathcal {O}}}}$ or ${\hat{o}}_{\ell }=C(V_\ell ) \in {\widehat{{\mathcal {O}}}}$, such that $t_{j'} = \mathrm{arg\,max}\Big \{\; v_j \; | \; {z^{{\ell }}_j = 0}, j \in \varOmega \Big \}=\bar{j}_\ell$. Hence, $t_{j'}\in V_\ell {\setminus } V_i$. We have $\min \Big \{ \max _{j \in V_{i}} v_j, \; \max _{j \in V_{\ell }} v_j \Big \} =\min \{v_{\bar{j}_i},v_{t_{j'}}\}= v_{ t_{j'}}=\max _{j \in V_{\ell } } v_j >\max _{j \in (V_{i} \cap V_{\ell } )} v_j,$ where in the equations we have used respectively the definitions of $\bar{j}_i$ and $\bar{j}_\ell$ along with $t_{j'}=\bar{j}_\ell$, the fact that $v_{t_{j'}} \le v_{\bar{j}_{i}}$. Whenever $V_i\cap V_\ell =\emptyset$, the strict inequality follows from $v_{t_{j'}}>0$ and our convention that $\max _{j\in V} v_j = 0$ for $V=\emptyset$. Whenever $V_i\cap V_\ell \ne \emptyset$, recall that if ${\hat{o}}_{i}\in \{B(V_i), C(V_i)\}$ is in ${\widehat{{\mathcal {O}}}}$ and ${\hat{o}}_{\ell }\in \{B(V_\ell ),C(V_\ell )\}$ is in ${\widehat{{\mathcal {O}}}}$, then from the premise of the lemma, we have $V_\kappa :=V_i\cap V_\ell$ is such that ${\hat{o}}_{\kappa }\in \{B(V_\kappa ),C(V_\kappa )\}$ is also in ${\widehat{{\mathcal {O}}}}$ which implies that the strict inequality above follows from $\bar{j}_\ell = t_{j'} \not \in V_i \cap V_\ell$, hence $\bar{j}_\kappa =t_{k_\kappa }$ for some $\rho \ge k_\kappa \ge j' + 1$ and that $v_{t_{j'}} > v_{t_{j' + 1}}\ge v_{\bar{j}_\kappa }$. Consequently, we reach a contradiction because this inequality implies $\psi (\mathbf {v}, V_i,V_\ell )>0$, which contradicts the premise of the lemma. As a result, $t_{j'}$ cannot exist, i.e., $z^i_{t_j}=0$ for all $j=k_i+1,\dots ,\rho$ in inequality (27). Hence, inequality (10) for this choice of T must be tight at any solution ${\hat{o}}_i\in {\widehat{{\mathcal {O}}}}$ satisfying the premise of the lemma. $\square$