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Exact SDP reformulations of adjustable robust linear programs with box uncertainties under separable quadratic decision rules via SOS representations of non-negativity

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Abstract

In this paper we show that two-stage adjustable robust linear programs with affinely adjustable data in the face of box data uncertainties under separable quadratic decision rules admit exact semi-definite program (SDP) reformulations in the sense that they share the same optimal values and admit a one-to-one correspondence between the optimal solutions. This result allows adjustable robust solutions of these robust linear programs to be found by simply numerically solving their SDP reformulations. We achieve this result by first proving a special sum-of-squares representation of non-negativity of a separable non-convex quadratic function over box constraints. Our reformulation scheme is illustrated via numerical experiments by applying it to an inventory-production management problem with the demand uncertainty. They demonstrate that our separable quadratic decision rule method to two-stage decision-making performs better than the single-stage approach and is capable of solving the inventory production problem with a greater degree of uncertainty in the demand.

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Data availability statement

The data generated during the current study are available from the corresponding author on reasonable request.

Notes

  1. Note that, for the remainder of this paper, we now use the symbol \(\top \) to stand for the transpose of a vector or matrix. This is to avoid confusion with the number of periods in the planning horizon, T.

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Acknowledgements

The authors are grateful to the referee for his or her constructive comments and valuable suggestions which have contributed to the final version of the paper. They are also grateful to Dr Frans de Ruiter (Tilburg University, The Netherlands), for providing assistance to our numerical experiments.

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Research was supported by a research grant from Australian Research Council.

Appendix: Exact reformulations for the inventory production problem

Appendix: Exact reformulations for the inventory production problem

Standard Formulation In order to transform the problem (IP) into a standard matrix form, we further define:

$$\begin{aligned} p^t = \begin{bmatrix} p_1^t \\ \vdots \\ p_I^t \end{bmatrix},\quad p = \begin{bmatrix} p^1 \\ \vdots \\ p^T\end{bmatrix},\quad c^t = \begin{bmatrix} c_1^t \\ \vdots \\ c_I^t \end{bmatrix},\quad c = \begin{bmatrix} c^1 \\ \vdots \\ c^T \end{bmatrix}, \quad P^t = \begin{bmatrix} P_1^t \\ \vdots \\ P_I^t \end{bmatrix}, \quad P = \begin{bmatrix} P^1 \\ \vdots \\ P^T\end{bmatrix}, \quad { \kappa = \begin{bmatrix} \kappa _1 \\ \vdots \\ \kappa _I \end{bmatrix}.} \end{aligned}$$

After also introducing the auxiliary (here-and-now) variable F as per the methodology in the proof of Theorem 3.1, (IP) is equivalently rewritten as

$$\begin{aligned}&(IP) \min _{p,\,F} F \\&\quad \text {subject to } -F + c^\top p \le { 0,} \\&\quad -p \le { 0_{IT},} \\&\quad p {\le P,} \\&\quad \left( {\mathbf {1}}_T^\top \otimes \text {Id }_I\right) p \le {\kappa ,} \\&\quad \begin{bmatrix} {\mathbf {1}}_{It} \\ {0_{I(T-t)}}\end{bmatrix}^\top p \le V_{\text {max }} - v_0 + \begin{bmatrix} {\mathbf {1}}_t \\ {0_{T-t}} \end{bmatrix}^\top d,\quad t = 1,\dots ,{T,} \\&\quad -\begin{bmatrix} {\mathbf {1}}_{It} \\ { 0_{I(T-t)}}\end{bmatrix}^\top p \le -V_{\text {min }} + v_0 - \begin{bmatrix} {\mathbf {1}}_t \\ { 0_{T-t}} \end{bmatrix}^\top d,\quad t = 1,\dots ,{ T,} \end{aligned}$$

or in a standard form,

$$\begin{aligned}&(IP) \min _{p,\, F} F\\&\quad \text {subject to } aF + Bp \le g_0 + { Gd,} \end{aligned}$$

where

$$\begin{aligned} a&= \begin{bmatrix} -1 \\ { 0_{2IT + 2T + I}}\end{bmatrix}\in {\mathbb {R}}^{2IT + 2T + I + 1},&B = \begin{bmatrix} c^\top \\ -\text {Id }_{IT} \\ \text {Id }_{IT} \\ {\mathbf {1}}_T^\top \otimes \text {Id }_I \\ ({\mathbf {1}}_{T\times T})_{\nabla }\otimes {\mathbf {1}}_I^\top \\ -({\mathbf {1}}_{T\times T})_{\nabla }\otimes {\mathbf {1}}_I^\top \end{bmatrix}\in {\mathbb {R}}^{(2IT + 2T + I + 1)\times T}{,} \\ g_0&= \begin{bmatrix} { 0_{IT + 1}} \\ P \\ \kappa \\ {\mathbf {1}}_T(V_{\text {max }}-v_0) \\ -{\mathbf {1}}_T(V_{\text {min }}-v_0)\end{bmatrix}\in {\mathbb {R}}^{2IT + 2T + I 1},&G = \begin{bmatrix} { 0_{(2IT + I + 1)\times T}} \\ ({\mathbf {1}}_{T\times T})_\nabla \\ -({\mathbf {1}}_{T\times T})_\nabla \end{bmatrix}\in {\mathbb {R}}^{(2IT + 2T + I + 1)\times T}{,} \end{aligned}$$

where \(\text {Id }_n\) is the \(n\times n\) identity matrix, \(\otimes \) refers to the Kronecker product, and \(M_\nabla \) is the lower triangular part of matrix M.

Recall we wish to solve \((IP-ARO)\) via a separable QDR of the form

$$\begin{aligned} p_i^t(d_{{\mathcal {I}}_t}) = p_i^t(d) = y_i^t + (w_i^t)^\top d + { d^\top Q_i^t d,} \end{aligned}$$

where \(w_i^t\in {\mathbb {R}}^T, (w_i^t)_j = 0\) for \(j \ge t\), and \(Q_i^t\in {\mathbb {R}}^{T\times T}, Q_i^t = \text {diag }( \xi _{i1}^t, \, \xi _{i2}^t, \, \dots \, , \xi _{i(t-1)}^t,\) \(\, 0, \, \dots \, , 0 )\). Let

$$\begin{aligned} y^t = \begin{bmatrix} y_1^t \\ \vdots \\ y_I^t\end{bmatrix},\quad y = \begin{bmatrix} y^1 \\ \vdots \\ y^T \end{bmatrix}, \quad W^t = \begin{bmatrix} (w_1^t)^\top \\ \vdots \\ (w_I^t)\top \end{bmatrix},\quad {W = \begin{bmatrix} W^1 \\ \vdots \\ W^T \end{bmatrix}.} \end{aligned}$$

Then \((IP-QDR)\) can be written as

$$\begin{aligned}&(IP-QDR) \min _{\begin{array}{c} y\in {\mathbb {R}}^{IT},\, W\in {\mathbb {R}}^{IT\times T} \\ Q_r\in {\mathbb {R}}^{T\times T},\, F\in {\mathbb {R}} \end{array}} F \\&\quad \text {subject to } aF + B\left( y + Wd + \begin{bmatrix} d^\top Q_1 d \\ \vdots \\ d^\top Q_{IT} d\end{bmatrix} \right) \le g_0 + Gd,\quad \forall { d\in {\mathcal {D}},} \end{aligned}$$

where \(W_{ij} = 0\) for \(j\ge \lceil \frac{i}{I}\rceil \), \(Q_{r} = Q_i^t\) for \(r = I(t-1)+i\) and \(Q_r = \text {diag }(\xi _r^{(1)},\dots ,\xi _r^{(t-1)},0,\dots ,0)\). The problem is now in a standard form. Notice that the coefficient matrix of the here-and-now variable F is also fixed, and has no affine dependence on the uncertainty d.

Exact SDP Reformulation Applying Theorem 3.1 to \((IP-QDR)\) we find that the exact SDP reformulation is

$$\begin{aligned}&(IP-SDP) \\&\quad \min _{\begin{array}{c} F\in {\mathbb {R}},\,\lambda _t^i\in {\mathbb {R}}, \\ y\in {\mathbb {R}}^{IT},\, W\in {\mathbb {R}}^{IT\times T} \\ Q_r\in {\mathbb {R}}^{T\times T},\, M^{i,t}\in {\mathbb {S}}^{3\times 3} \end{array}} F \\&\quad \text {subject to } M^{i,t} = \begin{bmatrix} M^{i,t}_{1,1} &{}\quad M^{i,t}_{1,2} &{}\quad M^{i,t}_{1,3} \\ M^{i,t}_{1,2} &{}\quad M^{i,t}_{2,2} &{}\quad M^{i,t}_{2,3} \\ M^{i,t}_{1,3} &{}\quad M^{i,t}_{2,3} &{}\quad M^{i,t}_{3,3} \end{bmatrix} \succeq 0,\\&\quad M^{i,t}_{1,1} = \lambda _t^i - (1-\delta )\left( W^\top b_i - g_i\right) _t d_t^* - { \sum _{r=1}^{IT}{(b_i)_r \xi _r^{(t)}\left( (1-\delta )d_t^*\right) ^2},} \\&\quad { M^{i,t}_{1,2} = 0,} \\&\quad 2M^{i,t}_{1,3} + M^{i,t}_{2,2}= 2\lambda _t^i - 2d_t^*\left( W^\top b_i - g_i\right) _t - 2\sum _{r=1}^{IT}{(b_i)_r \xi _r^{(t)}(1-\delta )(1+\delta ){d_t^*}^2}{,} \\&\quad { M^{i,t}_{2,3} = 0,} \\&M^{i,t}_{3,3} = \lambda _t^i - (1+\delta )\left( W^\top b_i - g_i\right) _td_t^* - \sum _{r=1}^{IT}{(b_i)_r\xi _r^{(t)}\left( (1+\delta )d_t^*\right) ^2}{,} \\&\quad \sum _{t=1}^{T}{\lambda _t^i} \le - \left( a_i F + b_i^\top y - (g_0)_i\right) {,} \\&\quad \text {for } i = 1,\dots ,m\quad { t = 1,\dots ,T.} \end{aligned}$$

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Chuong, T.D., Jeyakumar, V., Li, G. et al. Exact SDP reformulations of adjustable robust linear programs with box uncertainties under separable quadratic decision rules via SOS representations of non-negativity. J Glob Optim 81, 1095–1117 (2021). https://doi.org/10.1007/s10898-021-01050-x

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