arXiv subjects

Alain Billionnet

Publications and source records attributed to Alain Billionnet.

3 recordsLinked to original sources

arXiv2026

Efficient modeling of chemotherapy regimens using mixed-integer linear programming

In this article, we focus on determining a minimum-cost treatment program aimed at maintaining the size of a cancerous tumor at a level that allows the patient to live comfortably. At each predetermined point in a treatment horizon, the patient either receives drug treatment or does not. In the first case, the tumor shrinks and its size is multiplied by a constant factor lower than 1; in the second, it grows following an exponential or Gompertz growth law. We first demonstrate that a simple heuristic solution provides an optimal treatment program. We then show that the Gompertz function can be described, like the exponential function, by a simple recurrence relation that does not explicitly depend on time. Thanks to the characteristics of the logarithmic function, this property allows us to formulate the problem as a mixed-integer linear program. This result makes it possible to solve the problem very efficiently using one of the many solvers available to handle this type of program, and above all to consider and solve several extensions to the problem. In particular, we show how to determine which of the optimal equivalent solutions to the initial problem are the most relevant. We also show how to measure the effect of a marginal increase in the treatment budget on patient quality of life. Numerous computational experiments are presented to illustrate these issues.

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arXiv2022

Quantifying extinction probabilities of endangered species for phylogenetic conservation prioritization may not be as sensitive as might be feared

In this study we are concerned with the general problem of choosing from a set of endangered species T a subset S of k species to protect as a priority. Here, the interest to protect the species of S is assessed by the resulting expected phylogenetic diversity (ePD) of the set T, a widely used criterion for measuring the expected amount of evolutionary history associated with T. We consider that the survival of the protected species is assured and, on the contrary, that there is a risk of extinction for the unprotected species. The problem is easy to solve by a greedy type method if the extinction probabilities of the unprotected species are known but these probabilities are generally not easy to quantify. We show in this note that the choice of the precise values attributed to the extinction probabilities-provided it respects the rank of imperilment of each species-is not as decisive as might be feared for the considered problem. The values of these probabilities have a clear impact on the selection of the species to be protected but a little impact on the resulting ePD. More precisely, if T1 and T2 are the two optimal subsets of species corresponding to two scenarios (two different sets of probabilities) the ePDs of T1 and T2 , calculated with the probabilities of the first scenario-or with the probabilities of the second scenario-are not very different.

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arXiv2016

Using a conic bundle method to accelerate both phases of a quadratic convex reformulation

We present algorithm MIQCR-CB that is an advancement of method MIQCR~(Billionnet, Elloumi and Lambert, 2012). MIQCR is a method for solving mixed-integer quadratic programs and works in two phases: the first phase determines an equivalent quadratic formulation with a convex objective function by solving a semidefinite problem $(SDP)$, and, in the second phase, the equivalent formulation is solved by a standard solver. As the reformulation relies on the solution of a large-scale semidefinite program, it is not tractable by existing semidefinite solvers, already for medium sized problems. To surmount this difficulty, we present in MIQCR-CB a subgradient algorithm within a Lagrangian duality framework for solving $(SDP)$ that substantially speeds up the first phase. Moreover, this algorithm leads to a reformulated problem of smaller size than the one obtained by the original MIQCR method which results in a shorter time for solving the second phase. We present extensive computational results to show the efficiency of our algorithm.

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