Inverse optimization for multi-objective linear programming

Optimization of Water and Energy Spatial Patterns in the Cascade Pump Station Irrigation District

Cascade pump station irrigation districts (CPSIDs) consume large quantities of water and energy. Water- and energy-saving results and income increases are guaranteed under the sustainable development of the CPSID. The CPSID is divided into several sub-districts based on the elevation difference of topography and pump station distributions. The spatial patterns of crops and irrigation technologies can be changed by adjusting crop planting structures and developing drip irrigation in each sub-district. Its optimization will change the spatial patterns of irrigation water and energy consumption to achieve water- and energy-saving results, increase income, and provide an ecological advantage. To obtain the optimal spatial patterns of water and energy in the CPSID, a multi-objective linear programming model of minimum irrigation water consumption, minimum energy consumption, and highest crop output value was established. This model was applied to the Jingdian Phase I Irrigation District in northwest China, and an optimal scheme of water and energy spatial patterns was obtained. Compared with the present situation, the optimal scheme could save water by 26.18%, save energy by 29.38%, and increase income by 29.55%. The increased investment in the drip irrigation project would lead to reduced irrigation water and energy consumption and increased crop output value. The research results provide a scientific basis for the sustainable development of agriculture and ecological environment protection in the CPSID.

Imputing radiobiological parameters of the linear-quadratic dose-response model from a radiotherapy fractionation plan

The objective in cancer radiotherapy is to maximize tumor-kill while limiting toxic effects of radiation dose on nearby organs-at-risk (OAR). Given a fixed number of treatment sessions, planners thus face the problem of finding a dosing sequence that achieves this goal. This is called the fractionation problem, and has received steady attention over a long history in the clinical literature. Mathematical formulations of the resulting optimization problem utilize the linear-quadratic (LQ) framework to characterize radiation dose-response of tumors and OAR. This yields a nonconvex quadratically constrained quadratic program. The optimal dosing plan in this forward problem crucially depends on the parameters of the LQ model. Unfortunately, these parameters are difficult to estimate via in vitro or in vivo studies, and as such, their values are unknown to treatment planners. The clinical literature is thus replete with debates about what parameter values will make specific dosing plans effective. This paper formulates this as an inverse optimization problem. The LQ dose-response parameters appear in the objective function, the left hand side, and the right hand side of the forward problem, and none of the existing generic methods can provide an exact solution of the inverse problem. This paper exploits the structure of the problem and identifies all possible parameter values that render the given dosing plan optimal, in closed-form. This closed-form formula is applied to dosing-plans from three clinical studies published within the last two years.