A real moment-HSOS hierarchy for complex polynomial optimization with real coefficients

This paper proposes a real moment-HSOS hierarchy for complex polynomial optimization problems with real coefficients. We show that this hierarchy provides the same sequence of lower bounds as the complex analogue, yet is much cheaper to solve. In addition, we prove that global optimality is achieved when the ranks of the moment matrix and certain … Read more

Convex envelopes of bounded monomials on two-variable cones

\(\) We consider an \(n\)-variate monomial function that is restricted both in value by lower and upper bounds and in domain by two homogeneous linear inequalities. Such functions are building blocks of several problems found in practical applications, and that fall under the class of Mixed Integer Nonlinear Optimization. We show that the upper envelope … Read more

AI Hilbert: A New Paradigm for Scientific Discovery by Unifying Data and Background Knowledge

The discovery of scientific formulae that parsimoniously explain natural phenomena and align with existing background theory is a key goal in science. Historically, scientists have derived natural laws by manipulating equations based on existing knowledge, forming new equations, and verifying them experimentally. In recent years, data-driven scientific discovery has emerged as a viable competitor in settings with … Read more

Further Development in Convex Conic Reformulation of Geometric Nonconvex Conic Optimization Problems

\(\) A geometric nonconvex conic optimization problem (COP) was recently proposed by Kim, Kojima and Toh asa unified framework for convex conic reformulation of a class of quadratic optimization problems and polynomial optimization problems. The nonconvex COP minimizes a linear function over the intersection of a nonconvex cone K, a convex subcone J of the … Read more

On Common-Random-Numbers and the Complexity of Adaptive Sampling Trust-Region Methods

\(\) In the context of simulation optimization (SO), Common Random Numbers (CRN) is the practice of querying the simulation-based oracle with the same random number stream at each point visited by an SO algorithm. This practice is widely believed to facilitate SO algorithm efficiency by preserving structure inherent to the objective function and gradient sample-paths. … Read more

Exploring Nonlinear Kernels for Lipschitz Constant Estimation in Lower Bound Construction for Global Optimization

Bounds play a crucial role in guiding optimization algorithms, improving their speed and quality and providing optimality gaps. While Lipschitz constant-based lower bound construction is an effective technique, the quality of the linear bounds depends on the function’s topological properties. In this research, we improve upon this by incorporating nonlinear kernels and surrogate approximations to … Read more

A more efficient reformulation of complex SDP as real SDP

This note proposes a novel reformulation of complex semidefinite programs (SDPs) as real SDPs. As an application, we present an economical reformulation of complex SDP relaxations of complex polynomial optimization problems as real SDPs and derive some further reductions by exploiting structure of the complex SDP relaxations. Various numerical examples demonstrate that our new reformulation … Read more

Cutting planes from the simplex tableau for quadratically constrained optimization problems

We describe a method to generate cutting planes for quadratically constrained optimization problems. The method uses information from the simplex tableau of a linear relaxation of the problem in combination with McCormick estimators. The method is guaranteed to cut off a basic feasible solution of the linear relaxation that violates the quadratic constraints in the … Read more

Diagonal Partitioning Strategy Using Bisection of Rectangles and a Novel Sampling Scheme

In this paper we consider a global optimization problem, where the objective function is supposed to be Lipschitz-continuous with an unknown Lipschitz constant. Based on the recently introduced BIRECT (BIsection of RECTangles) algorithm, a new diagonal partitioning and sampling scheme is introduced. 0ur framework, called BIRECT-V (where V stands for vertices), combines bisection with sampling … Read more

A novel UCB-based batch strategy for Bayesian optimization

The optimization of expensive black-box functions appears in many situations. Bayesian optimization methods have been successfully applied to solve these prob- lems using well-known single-point acquisition functions. Nowadays, the develop- ments in technology allow us to perform evaluations of some of these expensive function in parallel. Therefore, there is a need for batch infill criteria … Read more