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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The data generated during the current study are available from the corresponding author on reasonable request.
Notes
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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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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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:
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
or in a standard form,
where
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
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
Then \((IP-QDR)\) can be written as
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
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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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DOI: https://doi.org/10.1007/s10898-021-01050-x
Keywords
- Adjustable robust linear optimization
- Semi-definite programs
- Sum of squares representations
- Nonconvex quadratic systems
- Quadratic decision rules