This paper mainly addresses the Monge mass transfer problem in the 1-D case. Through an ingenious approximation mechanism, one transforms the Monge problem into a sequence of minimization problems, which can be converted into a sequence of nonlinear differential equations with constraints by variational method. The existence and uniqueness of the solution for each equation can be demonstrated by applying the canonical duality method. Moreover, the duality method gives a sequence of perfect dual maximization problems. In the final analysis, one constructs the approximation of optimal mapping for the Monge problem according to the theoretical results.
The paper is accompanying "A general Duality Theorem for the Monge-Kantorovich Transport Problem". We explain the methods used in this article in an elementary setting and present two examples complementing the results obtained therein.
It is well-known that duality in the Monge-Kantorovich transport problem holds true provided that the cost function $c:X\times Y\to [0,\infty]$ is lower semi-continuous or finitely valued, but it may fail otherwise. We present a suitable notion of \emph{rectificaton} $c_r$ of the cost $c$, so that the Monge-Kantorovich duality holds true replacing $c$ by $c_r$. In particular, passing from $c$ to $c_r$ only changes the value of the primal Monge-Kantorovich problem. Finally, the rectified function $c_r $ is lower semi-continuous as soon as $X$ and $Y$ are endowed with proper topologies, thus emp…
The duality theory of the Monge--Kantorovich transport problem is analyzed in a general setting. The spaces $X, Y$ are assumed to be polish and equipped with Borel probability measures $μ$ and $ν$. The transport cost function $c:X\times Y \to [0,\infty]$ is assumed to be Borel. Our main result states that in this setting there is no duality gap, provided the optimal transport problem is formulated in a suitably relaxed way. The relaxed transport problem is defined as the limiting cost of the partial transport of masses $1-\varepsilon$ from $(X,μ)$ to $(Y, ν)$, as $\varepsilon >0$ tends to zero…
We consider the Monge-Kantorovich transport problem in an abstract measure theoretic setting. Our main result states that duality holds if $c:X\times Y\to [0,\infty)$ is an arbitrary Borel measurable cost function on the product of Polish spaces $X,Y$. In the course of the proof we show how to relate a non - optimal transport plan to the optimal transport costs via a ``subsidy'' function and how to identify the dual optimizer. We also provide some examples showing the limitations of the duality relations.
We present a general method, based on conjugate duality, for solving a convex minimization problem without assuming unnecessary topological restrictions on the constraint set. It leads to dual equalities and characterizations of the minimizers without constraint qualification. As an example of application, the Monge-Kantorovich optimal transport problem is solved in great detail. In particular, the optimal transport plans are characterized without restriction. This characterization improves the already existing literature on the subject.
We shall present a measure theoretical approach for which together with the Kantorovich duality provide an efficient tool to study the optimal transport problem. Specifically, we study the support of optimal plans where the cost function does not satisfy the classical twist condition in the two marginal problem as well as in the multi-marginal case when twistedness is limited to certain subsets.
A remarkable connection between optimal design and Monge transport was initiated in the years 1997 in the context of the minimal elastic compliance problem and where the euclidean metric cost was naturally involved. In this paper we present different variants in optimal design of mechanical structures, in particular focusing on the optimal pre-stressed elastic membrane problem. We show that the underlying metric cost is associated with an unknown maximal monotone map which maximizes the Monge-Kantorovich distance between two measures. In parallel with the classical duality theory leading to ex…
In this short note we show, providing counterexamples, that the "two important theorems" in the recent paper [Y, Yuan, Global optimization solutions to a class of non-convex quadratic minimization problems with quadratic constraints, in Canonical Duality Theory, D.Y. Gao et al.\ (eds), (AMMA, volume 37), Springer, 2017] are false.
This paper presents a canonical duality theory for solving a general nonconvex constrained optimization problem within a unified framework to cover Lagrange multiplier method and KKT theory. It is proved that if both target function and constraints possess certain patterns necessary for modeling real systems, a perfect dual problem (without duality gap)can be obtained in a unified form with global optimality conditions provided. While the popular augmented Lagrangian method may produce more difficult nonconvex problems due to the nonlinearity of constraints.
We propose to solve large instances of the non-convex optimization problems reformulated with canonical duality theory. To this aim we propose an interior point potential reduction algorithm based on the solution of the primal-dual total complementarity (Lagrange) function. We establish the global convergence result for the algorithm under mild assumptions and demonstrate the method on instances of the Sensor Network Localization problem. Our numerical results are promising and show the possibility of devising efficient interior points methods for non-convex duality.
This paper presents a canonical d.c. (difference of canonical and convex functions) programming problem, which can be used to model general global optimization problems in complex systems. It shows that by using the canonical duality theory, a large class of nonconvex minimization problems can be equivalently converted to a unified concave maximization problem over a convex domain, which can be solved easily under certain conditions. Additionally, a detailed proof for triality theory is provided, which can be used to identify local extremal solutions. Applications are illustrated and open prob…
General nonconvex optimization problems are studied by using the canonical duality-triality theory. The triality theory is proved for sums of exponentials and quartic polynomials, which solved an open problem left in 2003. This theory can be used to find the global minimum and local extrema, which bridges a gap between global optimization and nonconvex mechanics. Detailed applications are illustrated by several examples.
This paper presents a canonical duality theory for solving nonconvex minimization problem of Rosenbrock function. Extensive numerical results show that this benchmark test problem can be solved precisely and efficiently to obtain global optimal solutions.
DY Gao solely or together with some of his collaborators applied his Canonical duality theory (CDT) for solving some quadratic optimization problems with quadratic constraints. Unfortunately, in almost all papers we read on CDT there are unclear definitions, non convincing arguments in the proofs, and even false results. The aim of this paper is to treat rigorously quadratic optimization problems by the method suggested by CDT and to compare what we get with the results obtained by DY Gao and his collaborators on this topic in several papers.
This paper mainly investigates the approximation of a global maximizer of the 1-D Monge-Kantorovich mass transfer problem through the approach of nonlinear differential equations with Dirichlet boundary. Using an approximation mechanism, the primal maximization problem can be transformed into a sequence of minimization problems. By applying the canonical duality theory, one is able to derive a sequence of analytic solutions for the minimization problems. In the final analysis, the convergence of the sequence to a global maximizer of the primal Monge-Kantorovich problem will be demonstrated.
A unified model is addressed for general optimization problems in multi-scale complex systems. Based on necessary conditions and basic principles in physics, the canonical duality-triality theory is presented in a precise way to include traditional duality theories and popular methods as special applications. Two conjectures on NP-hardness are discussed, which should play important roles for correctly understanding and efficiently solving challenging real-world problems. Applications are illustrated for both nonconvex continuous optimization and mixed integer nonlinear programming. Misundersta…
This paper demonstrates a mathematically correct and computationally powerful method for solving 3D topology optimization problems. This method is based on canonical duality theory (CDT) developed by Gao in nonconvex mechanics and global optimization. It shows that the so-called NP-hard knapsack problem in topology optimization can be solved deterministically in polynomial time via a canonical penalty-duality (CPD) method to obtain precise 0-1 global optimal solution at each volume evolution. The relation between this CPD method and Gao's pure complementary energy principle is revealed for the…
We propose an extension of the computational fluid mechanics approach to the Monge-Kantorovich mass transfer problem, which was developed by Benamou-Brenier. Our extension allows optimal transfer of unnormalized and unequal masses. We obtain a one-parameter family of simple modifications of the formulation in [4]. This leads us to a new Monge-Ampere type equation and a new Kantorovich duality formula. These can be solved efficiently by, for example, the Chambolle-Pock primal-dual algorithm. This solution to the extended mass transfer problem gives us a simple metric for computing the distance …
We study a canonical duality method to solve a mixed-integer nonconvex fourth-order polynomial minimization problem with fixed cost terms. This constrained nonconvex problem can be transformed into a continuous concave maximization dual problem without duality gap. The global optimality conditions are proposed and the existence and uniqueness criteria are discussed. Application to a decoupled mixed-integer problem is illustrated and analytic solution for a global minimum is obtained under some suitable conditions. Several examples are given to show the method is effective.
This paper presents an application of Canonical duality theory to the solution of contact problems with Coulomb friction. The contact problem is formulated as a quasi-variational inequality which solution is found by solving its Karush-Kunt-Tucker system of equations. The complementarity conditions are reformulated by using the Fischer-Burmeister complementarity function, obtaining a non-convex global optimization problem. Then canonical duality theory is applied to reformulate the non-convex global optimization problem and define its optimality conditions, finding a solution of the original q…
The canonical duality theory has provided with a unified analytic solution to a range of discrete and continuous problems in global optimization, which can transform a nonconvex primal problem to a concave maximization dual problem over a convex domain without duality gap. This paper shows that under certain conditions, this canonical dual problem is equivalent to the standard semi-definite programming (SDP) problem, which can be solved by well-developed software packages. In order to avoid certain difficulties of using the SDP method, four strategies are proposed based on unconstrained approa…
One revisits the standard saddle-point method based on conjugate duality for solving convex minimization problems. Our aim is to reduce or remove unnecessary topological restrictions on the constraint set. Dual equalities and characterizations of the minimizers are obtained with weak or without constraint qualifications. The main idea is to work with intrinsic topologies which reflect some geometry of the objective function. The abstract results of this article are applied in other papers to the Monge-Kantorovich optimal transport problem and the minimization of entropy functionals.
This paper revisits the well-studied fixed point problem from a unified viewpoint of mathematical modeling and canonical duality theory, i.e. the original problem is first reformulated as a nonconvex optimization problem, its well-posedness is discussed based on objectivity principle in continuum physics; then the canonical duality theory is applied for solving this problem to obtain not only all fixed points, but also their stability properties. Applications are illustrated by challenging problems governed by nonconvex polynomial, exponential, and logarithmic operators. This paper shows that …
This paper presents global optimal solutions to a nonconvex quadratic minimization problem over a sphere constraint. The problem is well-known as a trust region subproblem and has been studied extensively for decades. The main challenge is the so called 'hard case', i.e., the problem has multiple solutions on the boundary of the sphere. By canonical duality theory, this challenging problem is able to reformed as an one-dimensional canonical dual problem without duality gap. Sufficient and necessary conditions are obtained by the triality theory, which can be used to identify whether the proble…
A novel canonical duality theory (CDT) is presented for solving general bilevel mixed integer nonlinear optimization governed by linear and quadratic knapsack problems. It shows that the challenging knapsack problems can be solved analytically in term of their canonical dual solutions. The existence and uniqueness of these analytical solutions are proved. NP-Hardness of the knapsack problems is discussed. A powerful CDT algorithm combined with an alternative iteration and a volume reduction method is proposed for solving the NP-hard bilevel knapsack problems. Application is illustrated by a be…
This paper presents a canonical duality approach for solving a general topology optimization problem of nonlinear elastic structures. By using finite element method, this most challenging problem can be formulated as a mixed integer nonlinear programming problem (MINLP), i.e. for a given deformation, the first-level optimization is a typical linear constrained 0-1 programming problem, while for a given structure, the second-level optimization is a general nonlinear continuous minimization problem in computational nonlinear elasticity. It is discovered that for linear elastic structures, first-…
Topology optimization for general materials is correctly formulated as a bi-level knapsack problem, which is considered to be NP-hard in global optimization and computer science. By using canonical duality theory (CDT) developed by the author, the linear knapsack problem can be solved analytically to obtain global optimal solution at each design iteration. Both uniqueness, existence, and NP-hardness are discussed. The novel CDT method for general topology optimization is refined and tested by both 2-D and 3-D benchmark problems. Numerical results show that without using filter and any other ar…
This paper presents a detailed proof of the triality theorem for a class of fourth-order polynomial optimization problems. The method is based on linear algebra but it solves an open problem on the double-min duality left in 2003. Results show that the triality theory holds strongly in a tri-duality form if the primal problem and its canonical dual have the same dimension; otherwise, both the canonical min-max duality and the double-max duality still hold strongly, but the double-min duality holds weakly in a symmetrical form. Four numerical examples are presented to illustrate that this theor…
This paper presents a canonical dual approach for solving nonconvex quadratic minimization problem. By using the canonical duality theory, nonconvex primal minimization problems over n-dimensional Lorentz cone can be transformed into certain canonical dual problems with only one dual variable, which can be solved by using standard convex minimization methods. Extremality conditions of these solutions are classified by the triality theory. Applications are illustrated.
Two probability distributions $μ$ and $ν$ in second stochastic order can be coupled by a supermartingale, and in fact by many. Is there a canonical choice? We construct and investigate two couplings which arise as optimizers for constrained Monge-Kantorovich optimal transport problems where only supermartingales are allowed as transports. Much like the Hoeffding-Fréchet coupling of classical transport and its symmetric counterpart, the antitone coupling, these can be characterized by order-theoretic minimality properties, as simultaneous optimal transports for certain classes of reward (or cos…
In this paper, we provide an alternative proof of the monotonicity principle for the optimal Skorokhod embedding problem established by Beiglböck, Cox and Huesmann. This principle presents a geometric characterization that reflects the desired optimality properties of Skorokhod embeddings. Our proof is based on the adaptation of the Monge-Kantorovich duality in our context together with a delicate application of the optional cross-section theorem and a clever conditioning argument.
We study the free probabilistic analog of optimal couplings for the quadratic cost, where classical probability spaces are replaced by tracial von Neumann algebras, and probability measures on $\mathbb{R}^m$ are replaced by non-commutative laws of $m$-tuples. We prove an analog of the Monge-Kantorovich duality which characterizes optimal couplings of non-commutative laws with respect to Biane and Voiculescu's non-commutative $L^2$-Wasserstein distance using a new type of convex functions. As a consequence, we show that if $(X,Y)$ is a pair of optimally coupled $m$-tuples of non-commutative ran…
Triality theory is proved for a general unconstrained global optimization problem. The method adopted is simple but mathematically rigorous. Results show that if the primal problem and its canonical dual have the same dimension, the triality theory holds strongly in the tri-duality form as it was originally proposed. Otherwise, both the canonical min-max duality and the double-max duality still hold strongly, but the double-min duality holds weakly in a super-symmetrical form as it was expected. Additionally, a complementary weak saddle min-max duality theorem is discovered. Therefore, an open…
An analogue of the quadratic Wasserstein (or Monge-Kantorovich) distance between Borel probability measures on $\mathbf{R}^d$ has been defined in [F. Golse, C. Mouhot, T. Paul: Commun. Math. Phys. 343 (2015), 165-205] for density operators on $L^2(\mathbf{R}^d)$, and used to estimate the convergence rate of various asymptotic theories in the context of quantum mechanics. The present work proves a Kantorovich type duality theorem for this quantum variant of the Monge-Kantorovich or Wasserstein distance, and discusses the structure of optimal quantum couplings. Specifically, we prove that optima…
We show that the deterministic past history of the Universe can be uniquely reconstructed from the knowledge of the present mass density field, the latter being inferred from the 3D distribution of luminous matter, assumed to be tracing the distribution of dark matter up to a known bias. Reconstruction ceases to be unique below those scales -- a few Mpc -- where multi-streaming becomes significant. Above 6 Mpc/h we propose and implement an effective Monge-Ampere-Kantorovich method of unique reconstruction. At such scales the Zel'dovich approximation is well satisfied and reconstruction becomes…
