Constraint Qualifications and Optimality Conditions for Nonconvex Semi-Infinite and Infinite Programs

The paper concerns the study of new classes of nonlinear and nonconvex optimization problems of the so-called infinite programming that are generally defined on infinite-dimensional spaces of decision variables and contain infinitely many of equality and inequality constraints with arbitrary (may not be compact) index sets. These problems reduce to semi-infinite programs in the case of finite-dimensional spaces of decision variables. We extend the classical Mangasarian-Fromovitz and Farkas-Minkowski constraint qualifications to such infinite and semi-infinite programs. The new qualification conditions are used for efficient computing the appropriate normal cones to sets of feasible solutions for these programs by employing advanced tools of variational analysis and generalized differentiation. In the further development we derive first-order necessary optimality conditions for infinite and semi-infinite programs, which are new in both finite-dimensional and infinite-dimensional settings.Comment: 28 page

Strong Metric (Sub)regularity of KKT Mappings for Piecewise Linear-Quadratic Convex-Composite Optimization

This work concerns the local convergence theory of Newton and quasi-Newton methods for convex-composite optimization: minimize f(x):=h(c(x)), where h is an infinite-valued proper convex function and c is C^2-smooth. We focus on the case where h is infinite-valued piecewise linear-quadratic and convex. Such problems include nonlinear programming, mini-max optimization, estimation of nonlinear dynamics with non-Gaussian noise as well as many modern approaches to large-scale data analysis and machine learning. Our approach embeds the optimality conditions for convex-composite optimization problems into a generalized equation. We establish conditions for strong metric subregularity and strong metric regularity of the corresponding set-valued mappings. This allows us to extend classical convergence of Newton and quasi-Newton methods to the broader class of non-finite valued piecewise linear-quadratic convex-composite optimization problems. In particular we establish local quadratic convergence of the Newton method under conditions that parallel those in nonlinear programming when h is non-finite valued piecewise linear

The use of Grossone in Mathematical Programming and Operations Research

The concepts of infinity and infinitesimal in mathematics date back to anciens Greek and have always attracted great attention. Very recently, a new methodology has been proposed by Sergeyev for performing calculations with infinite and infinitesimal quantities, by introducing an infinite unit of measure expressed by the numeral grossone. An important characteristic of this novel approach is its attention to numerical aspects. In this paper we will present some possible applications and use of grossone in Operations Research and Mathematical Programming. In particular, we will show how the use of grossone can be beneficial in anti--cycling procedure for the well-known simplex method for solving Linear Programming Problems and in defining exact differentiable Penalty Functions in Nonlinear Programming

On generalized semi-infinite optimization and bilevel optimization

The paper studies the connections and differences between bilevel problems (BL) and generalized semi-infinite problems (GSIP). Under natural assumptions (GSIP) can be seen as a special case of a (BL). We consider the so-called reduction approach for (BL) and (GSIP) leading to optimality conditions and Newton-type methods for solving the problems. We show by a structural analysis that for (GSIP)-problems the regularity assumptions for the reduction approach can be expected to hold generically at a solution but for general (BL)-problems not. The genericity behavior of (BL) and (GSIP) is in particular studied for linear problems

Infinite horizon sparse optimal control

A class of infinite horizon optimal control problems involving $L^p$-type cost functionals with $0<p\leq 1$ is discussed. The existence of optimal controls is studied for both the convex case with $p=1$ and the nonconvex case with $0<p<1$, and the sparsity structure of the optimal controls promoted by the $L^p$-type penalties is analyzed. A dynamic programming approach is proposed to numerically approximate the corresponding sparse optimal controllers

Successive Convexification of Non-Convex Optimal Control Problems and Its Convergence Properties

This paper presents an algorithm to solve non-convex optimal control problems, where non-convexity can arise from nonlinear dynamics, and non-convex state and control constraints. This paper assumes that the state and control constraints are already convex or convexified, the proposed algorithm convexifies the nonlinear dynamics, via a linearization, in a successive manner. Thus at each succession, a convex optimal control subproblem is solved. Since the dynamics are linearized and other constraints are convex, after a discretization, the subproblem can be expressed as a finite dimensional convex programming subproblem. Since convex optimization problems can be solved very efficiently, especially with custom solvers, this subproblem can be solved in time-critical applications, such as real-time path planning for autonomous vehicles. Several safe-guarding techniques are incorporated into the algorithm, namely virtual control and trust regions, which add another layer of algorithmic robustness. A convergence analysis is presented in continuous- time setting. By doing so, our convergence results will be independent from any numerical schemes used for discretization. Numerical simulations are performed for an illustrative trajectory optimization example.Comment: Updates: corrected wordings for LICQ. This is the full version. A brief version of this paper is published in 2016 IEEE 55th Conference on Decision and Control (CDC). http://ieeexplore.ieee.org/document/7798816