Efficient certification of feasibility and objective value of linear programs and its applications

The use of linear programming in various areas has increased with the significant improvement of specialized solvers. Linear programs are used as such to model practical problems, or as subroutines in algorithms such as formal proofs or branch-and-cut frameworks. In many situations a certified answer is needed, for example the guarantee that the linear program is feasible or infeasible, or a provably safe bound on its objective value. Most of the available solvers work with floating-point arithmetic and are thus subject to its shortcomings such as rounding errors or underflow, therefore they can deliver incorrect answers. While adequate for some applications, this is unacceptable for critical applications like flight controlling or nuclear plant management due to the potential catastrophic consequences. We propose a method that gives a certified answer whether a linear program is feasible or infeasible, or returns unknown'. The advantage of our method is that it is reasonably fast and rarely answers unknown'. It works by computing a safe solution that is in some way the best possible in the relative interior of the feasible set. To certify the relative interior, we employ exact arithmetic, whose use is nevertheless limited in general to critical places, allowing us to rnremain computationally efficient. Moreover, when certain conditions are fulfilled, our method is able to deliver a provable bound on the objective value of the linear program. We test our algorithm on typical benchmark sets and obtain higher rates of success compared to previous approaches for this problem, while keeping the running times acceptably small. The computed objective value bounds are in most of the cases very close to the known exact objective values. We prove the usability of the method we developed by additionally employing a variant of it in a different scenario, namely to improve the results of a Satisfiability Modulo Theories solver. Our method is used as a black box in the nodes of a branch-and-bound tree to implement conflict learning based on the certificate of infeasibility for linear programs consisting of subsets of linear constraints. The generated conflict clauses are in general small and give good rnprospects for reducing the search space. Compared to other methods we obtain significant improvements in the running time, especially on the large instances.

Imiquimod based transcutaneous immunization – insights and novel concepts

This work aims at developing a transcutaneous immunization (TCI) approach in order to activate cytotoxic T-cells. A tumor specific immune response was therefore generated by the TLR7-Agonist imiquimod. Five commercially available creams including the innovators product Aldara® 5% creme were assessed to ascertain their capability to induce an immune response in C57BL/6 mice after dermal administration. Moreover, creams were investigated regarding their imiquimod permeation in a Franz-diffusion cell model. Results obtained from this study were used to develop novel formulation approaches based on dissolved state imiquimod in a submicron scale range. High pressure homogenization ensured emulsification as well as particle size reduction. A freeze dried spreadable solid nanoemulsion based on sucrose fatty acid esters and oil components represented a major formulation approach. Within the scope of this approach the influence of pharmaceutical oils i.e. middle chain triglycerides, avocado oil, jojoba wax, and squalen was assessed towards their TCI performance. Furthermore, an aqueous jojoba wax based emulsion gel was developed. Unlike the innovators product, all formulations demonstrated a distinctly reduced imiquimod permeation across murine skin, a fact particularly evident in case of jojoba wax. Squalen significantly augmented in vivo immune response (p≤0.05 Mann-Whitney-Test). The emulsion gel demonstrated a 10fold decrease of imiquimod permeation. In comparison with the innovators product, the emulsion gel induced an equal immune response with a simultaneously enhanced tumor rejection in a mouse model.