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Register a new userApproximations for Pareto and Proper Pareto solutions and their KKT conditions
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In this article, we view the approximate version of Pareto and weak Pareto solutions of the multiobjective optimization problem through the lens of KKT type conditions. We also focus on an improved version of Geoffrion proper Pareto solutions and characterize them through saddle point and KKT type conditions. We present an approximate version of the improved Geoffrion proper solutions and propose our results in general settings.
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In this paper, we are interested in the existence of Pareto solutions to vector polynomial optimization problems over a basic closed semi-algebraic set. By invoking some powerful tools from real semi-algebraic geometry, we first introduce the concept called {it tangency varieties}; then we establish connections of the Palais--Smale condition, Cerami condition, {it M}-tameness, and properness related to the considered problem, in which the condition of regularity at infinity plays an essential role in deriving these connections. According to the obtained connections, we provide some sufficient conditions for existence of Pareto solutions to the problem in consideration, and we also give some examples to illustrate our main findings.
We settle the computational complexity of fundamental questions related to multicriteria integer linear programs, when the dimensions of the strategy space and of the outcome space are considered fixed constants. In particular we construct: 1. polynomial-time algorithms to exactly determine the number of Pareto optima and Pareto strategies; 2. a polynomial-space polynomial-delay prescribed-order enumeration algorithm for arbitrary projections of the Pareto set; 3. an algorithm to minimize the distance of a Pareto optimum from a prescribed comparison point with respect to arbitrary polyhedral norms; 4. a fully polynomial-time approximation scheme for the problem of minimizing the distance of a Pareto optimum from a prescribed comparison point with respect to the Euclidean norm.
Multi-Task Learning (MTL) is a well-established paradigm for training deep neural network models for multiple correlated tasks. Often the task objectives conflict, requiring trade-offs between them during model building. In such cases, MTL models can use gradient-based multi-objective optimization (MOO) to find one or more Pareto optimal solutions. A common requirement in MTL applications is to find an {it Exact} Pareto optimal (EPO) solution, which satisfies user preferences with respect to task-specific objective functions. Further, to improve model generalization, various constraints on the weights may need to be enforced during training. Addressing these requirements is challenging because it requires a search direction that allows descent not only towards the Pareto front but also towards the input preference, within the constraints imposed and in a manner that scales to high-dimensional gradients. We design and theoretically analyze such search directions and develop the first scalable algorithm, with theoretical guarantees of convergence, to find an EPO solution, including when box and equality constraints are imposed. Our unique method combines multiple gradient descent with carefully controlled ascent to traverse the Pareto front in a principled manner, making it robust to initialization. This also facilitates systematic exploration of the Pareto front, that we utilize to approximate the Pareto front for multi-criteria decision-making. Empirical results show that our algorithm outperforms competing methods on benchmark MTL datasets and MOO problems.
We present, (partially) analyze, and apply an efficient algorithm for the simulation of multivariate Pareto records. A key role is played by minima of the record-setting region (we call these generators) each time a new record is generated, and two highlights of our work are (i) efficient dynamic maintenance of the set of generators and (ii) asymptotic analysis of the expected number of generators at each time.
The Pareto model is very popular in risk management, since simple analytical formulas can be derived for financial downside risk measures (Value-at-Risk, Expected Shortfall) or reinsurance premiums and related quantities (Large Claim Index, Return Period). Nevertheless, in practice, distributions are (strictly) Pareto only in the tails, above (possible very) large threshold. Therefore, it could be interesting to take into account second order behavior to provide a better fit. In this article, we present how to go from a strict Pareto model to Pareto-type distributions. We discuss inference, and derive formulas for various measures and indices, and finally provide applications on insurance losses and financial risks.
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