Convex projection and convex multi-objective optimization

Convex Optimization without Projection Steps

We study the general problem of minimizing a convex function over a compact convex domain. We will investigate a simple iterative approximation algorithm based on the method by Frank & Wolfe [FW56], that does not need projection steps in order to stay inside the optimization domain. Instead of a projection step, the linearized problem defined by a current subgradient is solved, which gives a st...

The Fixed Point Property in Convex Multi-objective Optimization Problem

In this paper we study the Pareto-optimal solutions in convex multi-objective optimization with compact and convex feasible domain. One of the most important problems in multi-objective optimization is the investigation of the topological structure of the Pareto sets. We present the problem of construction of a retraction function of the feasible domain onto Paretooptimal set, if the objective ...

Non-Convex and Multi-Objective Optimization in Data Mining - Non-Convex and Multi-Objective Optimization for Statistical Learning and Numerical Feature Engineering

Multi-scale exploration of convex functions and bandit convex optimization

We construct a new map from a convex function to a distribution on its domain, with the property that this distribution is a multi-scale exploration of the function. We use this map to solve a decadeold open problem in adversarial bandit convex optimization by showing that the minimax regret for this problem is Õ(poly(n) √ T ), where n is the dimension and T the number of rounds. This bound is ...

Multi-Period Trading via Convex Optimization

We consider a basic model of multi-period trading, which can be used to evaluate the performance of a trading strategy. We describe a framework for single-period optimization, where the trades in each period are found by solving a convex optimization problem that trades off expected return, risk, transaction cost and holding cost such as the borrowing cost for shorting assets. We then describe ...