M. R. Hestenes London: John Wiley. 1967. Pp. xii + 405. Price £5. Professor Hestenes has been a leading researcher on optimization theory since the early 1930's and this book contains much of his original work in the field. The first three chapters are of an introductory nature and contain the classical, calculus of variations, approach of first and second variations.

Calculus of Variations and Optimal Control Theory

We use a Fenchel duality scheme for solving control-constrained linear-quadratic optimal control problems We derive the dual of the optimal control problem explicitly, where the control constraints are embedded in the dual objective functional, which turns out to be continuously differentiable We specifically prove that strong duality and saddle point properties hold We carry out numerical experiments with the discretized primal and dual formulations of the problem, for which we implement powerful existing finite-dimensional optimization techniques and associated software We illustrate that by solving the dual of the optimal control problem, instead of the primal one, significant computational savings can be achieved Other numerical advantages are also discussed

A duality approach for solving control-constrained linear-quadratic optimal control problems ∗

We introduce and study a new dual condition which characterizes zero
duality gap in nonsmooth convex optimization. We prove that our
condition is weaker than all existing constraint qualifications,
including the closed epigraph condition. Our dual condition was inspired by, and is weaker than, the
so-called Bertsekas' condition for monotropic programming problems. We
give several corollaries of our result and special cases as
applications. We pay special attention to the polyhedral and sublinear
cases, and their implications in convex optimization.

Conditions for zero duality gap in convex programming

The linear quadratic regulator problem is central in optimal control and was investigated since the very beginning of control theory. Nevertheless, when it includes affine state constraints, it remains very challenging from the classical ``maximum principle`` perspective. In this study we present how matrix-valued reproducing kernels allow for an alternative viewpoint. We show that the quadratic objective paired with the linear dynamics encode the relevant kernel, defining a Hilbert space of controlled trajectories. Drawing upon kernel formalism, we introduce a strengthened continuous-time convex optimization problem which can be tackled exactly with finite dimensional solvers, and which solution is interior to the constraints. When refining a time-discretization grid, this solution can be made arbitrarily close to the solution of the state-constrained Linear Quadratic Regulator. We illustrate the implementation of this method on a path-planning problem.

Linearly-constrained Linear Quadratic Regulator from the viewpoint of kernel methods.

An improved collaborative optimization for multidisciplinary problems with coupled design variables

Strong duality for generalized monotropic programming in infinite dimensions

In this paper, we aim to establish some important properties about preinvex and quasi preinvex fuzzy mappings based on ranking value function of fuzzy numbers proposed by Syau and Lee. We p...

Fuzzy preinvexity via ranking value functions with applications to fuzzy optimization problems

On semi strongly (E, F)-convex functions and semi strongly (E, F)-convex optimization problems

The class of quasi semi -convex functions and pseudo semi -convex functions are presented in this paper by combining the class of -convex functions with the class of quasi semi -convex functions and pseudo semi -convex functions, respectively. Various non-trivial examples are introduced to illustrate the new functions and show their relationships with -convex functions recently introduced in the literature. Different general properties and characteristics of this class of functions are established. In addition, some optimality properties of generalized non-linear optimization problems are discussed. In this generalized optimization problems, we used, as the objective function, quasi semi -convex (respectively, strictly quasi semi -convex functions and pseudo semi -convex functions), and the constraint set is -convex set.

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Saba Naser Majeed

Papers

A duality approach for solving control-constrained linear-quadratic optimal control problems ∗

Strong duality for generalized monotropic programming in infinite dimensions

Fuzzy preinvexity via ranking value functions with applications to fuzzy optimization problems

On semi strongly (E, F)-convex functions and semi strongly (E, F)-convex optimization problems

Quasi Semi and Pseudo Semi (p,E)-Convexity in Non-Linear Optimization Programming