Constrained Global Optimization Research Papers (original) (raw)
A summary of the reserch done in global optimization at the Numerical Analysis Departament of IIMAS-UNAM is given. The concept of the Tunnelling Function and the key ideas of the Tunnelling Algorithm as applied to Unconstrained Global... more
A summary of the reserch done in global optimization at the Numerical Analysis Departament of IIMAS-UNAM is given. The concept of the Tunnelling Function and the key ideas of the Tunnelling Algorithm as applied to Unconstrained Global Optimization, Stabilization of Newton's Method and Constrained Global Optimization are presented. Numerical results for several examples are given, they have from one to ten variables and from three to several thousands of local minima, clearly illustrating the robustness of the Tunnelling Algorithm.
This paper presents a canonical duality theory for solving a general nonconvex quadratic minimization problem with nonconvex constraints. By using the canonical dual transformation developed by the first author, the nonconvex primal... more
This paper presents a canonical duality theory for solving a general nonconvex quadratic minimization problem with nonconvex constraints. By using the canonical dual transformation developed by the first author, the nonconvex primal problem can be converted into a canonical dual problem with zero duality gap. A general analytical solution form is obtained. Both global and local extrema of the nonconvex problem can be identified by the triality theory associated with the canonical duality theory. Illustrative applications to quadratic minimization with multiple quadratic constraints, box/integer constraints, and general nonconvex polynomial constraints are discussed, along with insightful connections to classical Lagrangian duality. Criteria for the existence and uniqueness of optimal solutions are presented. Several numerical examples are provided.
We present a modification of the DIRECT (DIviding RECTangles) algorithm, called DIRECT-G, to solve a box-constrained global optimization problem arising in the detection of gravitational waves emitted by coalescing binary systems of... more
We present a modification of the DIRECT (DIviding RECTangles) algorithm, called DIRECT-G, to solve a box-constrained global optimization problem arising in the detection of gravitational waves emitted by coalescing binary systems of compact objects. This is a hard problem, since the objective function is highly nonlinear and expensive to evaluate, has a huge number of local extrema and unavailable derivatives. DIRECT performs a sampling of the feasible domain over a set of points that becomes dense in the limit, thus ensuring the ...
A summary of the reserch done in global optimization at the Numerical Analysis Departament of IIMAS-UNAM is given. The concept of the Tunnelling Function and the key ideas of the Tunnelling Algorithm as applied to Unconstrained Global... more
A summary of the reserch done in global optimization at the Numerical Analysis Departament of IIMAS-UNAM is given. The concept of the Tunnelling Function and the key ideas of the Tunnelling Algorithm as applied to Unconstrained Global Optimization, Stabilization of Newton's Method and Constrained Global Optimization are presented. Numerical results for several examples are given, they have from one to ten variables and from three to several thousands of local minima, clearly illustrating the robustness of the Tunnelling Algorithm.
Presented here is a global optimization algorithm to solve n-dimensional box-constrained non-convex global optimization problems. The algorithm is referred to as Multi-Point Moving Grid (MPMG) method. It is of derivative-free type and... more
Presented here is a global optimization algorithm to solve n-dimensional box-constrained non-convex global optimization problems. The algorithm is referred to as Multi-Point Moving Grid (MPMG) method. It is of derivative-free type and does not require an initial guess or initialization procedure. The method is always convergent to the global solution as long as the objective function is Lipschitz continuous along with the first order and second order optimality conditions are satisfied. With increase in dimension size the computational cost of the method linearly grows which shows the method is efficient. The method is as well accurate to the largest extent. Global optimization is often NP-hard when the dimension of problem is high. Deterministic global optimization methods yield accurate results but are quite expensive in cost when problem is high-dimensional. On the other hand, stochastic methods are efficient and fast but they do not guarantee the accuracy of the solution found. However, MPMG search method developed in this study is simultaneously efficient and accurate. The efficiency and accuracy of the method is shown through numerical exercise and by a concise mathematical proof. The method is successfully capable to accurately and efficiently spot on the global optimizer for an exponential function with 3000 dimensions. Also a purportedly NP-hard quadratic programming problem is easily solved by the MPMG method. Several other high-dimensional multi-modal bench-mark functions available in the existing literature are successfully examined and treated by the method. For low dimensional problems and on an ordinary PC the solution is a fraction of a second away from a click and also for high dimensional problems the solution is not far away. The convergence rate is linear with a 0.5 asymptotic error constant.
Elements and techniques of state-of-the-art automatically verified constrained global optimization algorithms are reviewed, including a description of ways of rigorously verifying feasibility for equality constraints and a careful... more
Elements and techniques of state-of-the-art automatically verified constrained global optimization algorithms are reviewed, including a description of ways of rigorously verifying feasibility for equality constraints and a careful consideration of the role of active inequality constraints. Previously developed algorithms and general work on the subject are also listed. Limitations of present knowledge are mentioned, and advice is given on which techniques to use in various contexts. Applications are discussed.
A formal integer programming (IP) model, which simultaneously schedules and allocates functional units, registers, and buses is presented for synthesizing cost-constrained globally optimal architectures. This research is important for... more
A formal integer programming (IP) model, which simultaneously schedules and allocates functional units, registers, and buses is presented for synthesizing cost-constrained globally optimal architectures. This research is important for industry by providing optimal schedules which minimize interconnect costs and interface to analog and asynchronous processes, since these are seen as key to synthesizing high-performance architectures. A partially structured tight IP formulation of the architectural synthesis problem provides globally optimal schedules for piecewise linear cost functions, using branch and bound, in execution times faster than previous research