Second-order optimality conditions for interval-valued functions

This work is included in the search of optimality conditions for solutions to the scalar interval optimization problem, both constrained and unconstrained, by means of second-order optimality conditions. As it is known, these conditions allow us to reject some candidates to minima that arise from the first-order conditions. We will define new concepts such as second-order gH-derivative for interval-valued functions, 2-critical points, and 2-KKT-critical points. We obtain and present new types of interval-valued functions, such as 2-pseudoinvex, characterized by the property that all their second-order stationary points are global minima. We extend the optimality criteria to the semi-infinite programming problem and obtain duality theorems. These results represent an improvement in the treatment of optimization problems with interval-valued functions.Funding for open access publishing: Universidad de Cádiz/CBUA. The research has been supported by MCIN through grant MCIN/AEI/PID2021-123051NB-I00

Optimal Predictive Eco-Driving Cycles for Conventional, Electric, and Hybrid Electric Cars

International audienceIn this paper, the computation of eco-driving cycles for electric, conventional and hybrid vehicles using receding horizon and optimal control is studied. The problem is formulated as consecutive-optimization problems aiming at minimizing the vehicle energy consumption under traffic and speed constraints. The impact of the look-ahead distance and the optimization frequency on the optimal speed computation is studied to find a trade-off between the optimality and the computation time of the algorithm. For the three architectures considered, simulation results show that in urban driving conditions, a look-ahead distance of 300m to 500m leads to a sub-optimality less than 1% in the energy consumption compared to the global solution. For highway driving conditions, a look-ahead distance of 1km to 1.5km leads to a sub-optimality less than 2% compared to the global solution

Global rates of convergence for nonconvex optimization on manifolds

We consider the minimization of a cost function $f$ on a manifold $M$ using Riemannian gradient descent and Riemannian trust regions (RTR). We focus on satisfying necessary optimality conditions within a tolerance $\varepsilon$. Specifically, we show that, under Lipschitz-type assumptions on the pullbacks of $f$ to the tangent spaces of $M$, both of these algorithms produce points with Riemannian gradient smaller than $\varepsilon$ in $O(1/\varepsilon^2)$ iterations. Furthermore, RTR returns a point where also the Riemannian Hessian's least eigenvalue is larger than $-\varepsilon$ in $O(1/\varepsilon^3)$ iterations. There are no assumptions on initialization. The rates match their (sharp) unconstrained counterparts as a function of the accuracy $\varepsilon$ (up to constants) and hence are sharp in that sense. These are the first deterministic results for global rates of convergence to approximate first- and second-order Karush-Kuhn-Tucker points on manifolds. They apply in particular for optimization constrained to compact submanifolds of $\mathbb{R}^n$, under simpler assumptions.Comment: 33 pages, IMA Journal of Numerical Analysis, 201

Optimal Linear Precoding Strategies for Wideband Non-Cooperative Systems based on Game Theory-Part II: Algorithms

In this two-part paper, we address the problem of finding the optimal precoding/multiplexing scheme for a set of non-cooperative links sharing the same physical resources, e.g., time and bandwidth. We consider two alternative optimization problems: P.1) the maximization of mutual information on each link, given constraints on the transmit power and spectral mask; and P.2) the maximization of the transmission rate on each link, using finite order constellations, under the same constraints as in P.1, plus a constraint on the maximum average error probability on each link. Aiming at finding decentralized strategies, we adopted as optimality criterion the achievement of a Nash equilibrium and thus we formulated both problems P.1 and P.2 as strategic noncooperative (matrix-valued) games. In Part I of this two-part paper, after deriving the optimal structure of the linear transceivers for both games, we provided a unified set of sufficient conditions that guarantee the uniqueness of the Nash equilibrium. In this Part II, we focus on the achievement of the equilibrium and propose alternative distributed iterative algorithms that solve both games. Specifically, the new proposed algorithms are the following: 1) the sequential and simultaneous iterative waterfilling based algorithms, incorporating spectral mask constraints; 2) the sequential and simultaneous gradient projection based algorithms, establishing an interesting link with variational inequality problems. Our main contribution is to provide sufficient conditions for the global convergence of all the proposed algorithms which, although derived under stronger constraints, incorporating for example spectral mask constraints, have a broader validity than the convergence conditions known in the current literature for the sequential iterative waterfilling algorithm.Comment: Paper submitted to IEEE Transactions on Signal Processing, February 22, 2006. Revised March 26, 2007. Accepted June 5, 2007. To appear on IEEE Transactions on Signal Processing, 200

Linear programming on the Stiefel manifold

Linear programming on the Stiefel manifold (LPS) is studied for the first time. It aims at minimizing a linear objective function over the set of all $p$-tuples of orthonormal vectors in ${\mathbb R}^n$ satisfying $k$ additional linear constraints. Despite the classical polynomial-time solvable case $k=0$, general (LPS) is NP-hard. According to the Shapiro-Barvinok-Pataki theorem, (LPS) admits an exact semidefinite programming (SDP) relaxation when $p(p+1)/2\le n-k$, which is tight when $p=1$. Surprisingly, we can greatly strengthen this sufficient exactness condition to $p\le n-k$, which covers the classical case $p\le n$ and $k=0$. Regarding (LPS) as a smooth nonlinear programming problem, we reveal a nice property that under the linear independence constraint qualification, the standard first- and second-order {\it local} necessary optimality conditions are sufficient for {\it global} optimality when $p+1\le n-k$

Inexactness of SDP Relaxation and Valid Inequalities for Optimal Power Flow

It has been recently proven that the semidefinite programming (SDP) relaxation of the optimal power flow problem over radial networks is exact under technical conditions such as not including generation lower bounds or allowing load over-satisfaction. In this paper, we investigate the situation where generation lower bounds are present. We show that even for a two-bus one-generator system, the SDP relaxation can have all possible approximation outcomes, that is (1) SDP relaxation may be exact or (2) SDP relaxation may be inexact or (3) SDP relaxation may be feasible while the OPF instance may be infeasible. We provide a complete characterization of when these three approximation outcomes occur and an analytical expression of the resulting optimality gap for this two-bus system. In order to facilitate further research, we design a library of instances over radial networks in which the SDP relaxation has positive optimality gap. Finally, we propose valid inequalities and variable bound tightening techniques that significantly improve the computational performance of a global optimization solver. Our work demonstrates the need of developing efficient global optimization methods for the solution of OPF even in the simple but fundamental case of radial networks

Nondifferentiable multiobjective programming problem under strongly K-Gf-pseudoinvexity assumptions

[EN] In this paper we consider the introduction of the concept of (strongly) K-G(f)-pseudoinvex functions which enable to study a pair of nondifferentiable K-G- Mond-Weir type symmetric multiobjective programming model under such assumptions.Dubey, R.; Mishra, LN.; Sánchez Ruiz, LM.; Sarwe, DU. (2020). Nondifferentiable multiobjective programming problem under strongly K-Gf-pseudoinvexity assumptions. Mathematics. 8(5):1-11. https://doi.org/10.3390/math8050738S11185Antczak, T. (2007). New optimality conditions and duality results of type in differentiable mathematical programming. Nonlinear Analysis: Theory, Methods & Applications, 66(7), 1617-1632. doi:10.1016/j.na.2006.02.013Antczak, T. (2008). On G-invex multiobjective programming. Part I. Optimality. Journal of Global Optimization, 43(1), 97-109. doi:10.1007/s10898-008-9299-5Ferrara, M., & Viorica-Stefanescu, M. (2008). Optimality conditions and duality in multiobjective programming with invexity. YUJOR, 18(2), 153-165. doi:10.2298/yjor0802153fChen, X. (2004). Higher-order symmetric duality in nondifferentiable multiobjective programming problems. Journal of Mathematical Analysis and Applications, 290(2), 423-435. doi:10.1016/j.jmaa.2003.10.004Long, X. (2013). Sufficiency and duality for nonsmooth multiobjective programming problems involving generalized univex functions. Journal of Systems Science and Complexity, 26(6), 1002-1018. doi:10.1007/s11424-013-1089-6Dubey, R., Mishra, L. N., & Sánchez Ruiz, L. M. (2019). Nondifferentiable G-Mond–Weir Type Multiobjective Symmetric Fractional Problem and Their Duality Theorems under Generalized Assumptions. Symmetry, 11(11), 1348. doi:10.3390/sym11111348Pitea, A., & Postolache, M. (2011). Duality theorems for a new class of multitime multiobjective variational problems. Journal of Global Optimization, 54(1), 47-58. doi:10.1007/s10898-011-9740-zPitea, A., & Antczak, T. (2014). Proper efficiency and duality for a new class of nonconvex multitime multiobjective variational problems. Journal of Inequalities and Applications, 2014(1). doi:10.1186/1029-242x-2014-333Dubey, R., Deepmala, & Narayan Mishra, V. (2020). Higher-order symmetric duality in nondifferentiable multiobjective fractional programming problem over cone contraints. Statistics, Optimization & Information Computing, 8(1), 187-205. doi:10.19139/soic-2310-5070-60