arXiv subjects

Jiawang Nie

Publications and source records attributed to Jiawang Nie.

At least 37 records · Page 2Linked to original sources

arXiv2024

The Rank-1 Completion Problem for Cubic Tensors

This paper studies the rank-$1$ tensor completion problem for cubic tensors. First of all, we show that this problem is equivalent to a special rank-$1$ matrix recovery problem. When the tensor is strongly rank-$1$ completable, we show that the problem is equivalent to a rank-$1$ matrix completion problem and it can be solved by an iterative formula. For other cases, we propose both nuclear norm relaxation and moment relaxation methods for solving the resulting rank-$1$ matrix recovery problem. The nuclear norm relaxation sometimes returns a rank-$1$ tensor completion, while sometimes it does not. When it fails, we apply the moment hierarchy of semidefinite programming relaxations to solve the rank-$1$ matrix recovery problem. The moment hierarchy can always get a rank-$1$ tensor completion, or detect its nonexistence. Numerical experiments are shown to demonstrate the efficiency of these proposed methods.

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arXiv2024

Finite convergence of Moment-SOS relaxations with non-real radical ideals

We consider the linear conic optimization problem with the cone of nonnegative polynomials. Its dual optimization problem is the generalized moment problem. Moment-SOS relaxations are powerful for solving them. This paper studies finite convergence of the Moment-SOS hierarchy when the constraining set is defined by equations whose ideal may not be real radical. Under the archimedeanness, we show that the Moment-SOS hierarchy has finite convergence if some classical optimality conditions hold at every minimizer of the optimal nonnegative polynomial for the linear conic optimization problem. When the archimedeanness fails (this is the case for unbounded sets), we propose a homogenized Moment-SOS hierarchy and prove its finite convergence under similar assumptions. Furthermore, we also prove the finite convergence of the Moment-SOS hierarchy with denominators. In particular, this paper resolves a conjecture posed in the earlier work.

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arXiv2024

Towards Global Solutions for Nonconvex Two-Stage Stochastic Programs: A Polynomial Lower Approximation Approach

This paper tackles the challenging problem of finding global optimal solutions for two-stage stochastic programs with continuous decision variables and nonconvex recourse functions. We introduce a two-phase approach. The first phase involves the construction of a polynomial lower bound for the recourse function through a linear optimization problem over a nonnegative polynomial cone. Given the complex structure of this cone, we employ semidefinite relaxations with quadratic modules to facilitate our computations. In the second phase, we solve a surrogate first-stage problem by substituting the original recourse function with the polynomial lower approximation obtained in the first phase. Our method is particularly advantageous for two reasons: it not only generates global lower bounds for the nonconvex stochastic program, aiding in the certificate of global optimality for prospective solutions like stationary solutions computed from other methods, but it also yields an explicit polynomial approximation for the recourse function through the solution of a linear conic optimization problem, where the number of variables is independent of the support of the underlying random vector. Therefore, our approach is particularly suitable for the case where the random vector follows a continuous distribution or when dealing with a large number of scenarios. Numerical experiments are conducted to demonstrate the effectiveness of our proposed approach.

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arXiv2024

Polynomial Optimization Over Unions of Sets

This paper studies the polynomial optimization problem whose feasible set is a union of several basic closed semialgebraic sets. We propose a unified hierarchy of Moment-SOS relaxations to solve it globally. Under some assumptions, we prove the asymptotic or finite convergence of the unified hierarchy. Special properties for the univariate case are discussed.The application for computing $(p,q)$-norms of matrices is also presented.

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arXiv2024

Moment-SOS relaxations for moment and tensor recovery problems

This paper studies moment and tensor recovery problems whose decomposing vectors are contained in some given semialgebraic sets. We propose Moment-SOS relaxations with generic objectives for recovering moments and tensors, whose decomposition lengths are expected to be low. This kind of problems have broad applications in various tensor decomposition questions. Numerical experiments are provided to demonstrate the efficiency of this approach.

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arXiv2024

Diagonal Gaussian Mixture Models and Higher Order Tensor Decompositions

This paper studies how to recover parameters in diagonal Gaussian mixture models using tensors. High-order moments of the Gaussian mixture model are estimated from samples. They form incomplete symmetric tensors generated by hidden parameters in the model. We propose to use generating polynomials to compute incomplete symmetric tensor approximations. The obtained decomposition is utilized to recover parameters in the model. We prove that our recovered parameters are accurate when the estimated moments are accurate. Using high-order moments enables our algorithm to learn Gaussian mixtures with more components. For a given model dimension and order, we provide an upper bound of the number of components in the Gaussian mixture model that our algorithm can compute.

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arXiv2023

The Multi-Objective Polynomial Optimization

The multi-objective optimization is to optimize several objective functions over a common feasible set. Since the objectives usually do not share a common optimizer, people often consider (weakly) Pareto points. This paper studies multi-objective optimization problems that are given by polynomial functions. First, we study the convex geometry for (weakly) Pareto values and give a convex representation for them. Linear scalarization problems (LSPs) and Chebyshev scalarization problems (CSPs) are typical approaches for getting (weakly) Pareto points. For LSPs, we show how to use tight relaxations to solve them, how to detect existence or nonexistence of proper weights. For CSPs, we show how to solve them by moment relaxations. Moreover, we show how to check if a given point is a (weakly) Pareto point or not and how to detect existence or nonexistence of (weakly) Pareto points. We also study how to detect unboundedness of polynomial optimization, which is used to detect nonexistence of proper weights or (weakly) Pareto points.

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arXiv2023

Nash Equilibrium Problems of Polynomials

This paper studies Nash equilibrium problems that are given by polynomial functions. We formulate efficient polynomial optimization problems for computing Nash equilibria. The Lasserre type Moment-SOS relaxations are used to solve them. Under generic assumptions, the method can find a Nash equilibrium if there is one. Moreover, it can find all Nash equilibria if there are finitely many ones of them. The method can also detect nonexistence if there is no Nash equilibrium.

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arXiv2023

Solving polynomial variational inequality problems via Lagrange multiplier expressions and Moment-SOS relaxations

In this paper, we study variational inequality problems (VIPs) with involved mappings and feasible sets characterized by polynomial functions (namely, polynomial VIPs). We propose a numerical algorithm for computing solutions to polynomial VIPs based on Lagrange multiplier expressions and the Moment-SOS hierarchy of semidefinite relaxations. We also extend our approach to finding more or even all solutions to polynomial VIPs. We show that the method proposed in this paper can find solutions or detect the nonexistence of solutions within finitely many steps, under some general assumptions. In addition, we show that if the VIP is given by generic polynomials, then it has finitely many Karush-Kuhn-Tucker points, and our method can solve it within finitely many steps. Numerical experiments are conducted to illustrate the efficiency of the proposed methods.

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arXiv2023

Rational Generalized Nash Equilibrium Problems

This paper studies generalized Nash equilibrium problems that are given by rational functions. The optimization problems are not assumed to be convex. Rational expressions for Lagrange multipliers and feasible extensions of KKT points are introduced to compute a generalized Nash equilibrium (GNE). We give a hierarchy of rational optimization problems to solve rational generalized Nash equilibrium problems. The existence and computation of feasible extensions are studied. The Moment-SOS relaxations are applied to solve the rational optimization problems. Under some general assumptions, we show that the proposed hierarchy can compute a GNE if it exists or detect its nonexistence. Numerical experiments are given to show the efficiency of the proposed method.

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arXiv2022

Distributionally Robust Optimization with Moment Ambiguity Sets

This paper studies distributionally robust optimization (DRO) when the ambiguity set is given by moments for the distributions. The objective and constraints are given by polynomials in decision variables. We reformulate the DRO with equivalent moment conic constraints. Under some general assumptions, we prove the DRO is equivalent to a linear optimization problem with moment and psd polynomial cones. A Moment-SOS relaxation method is proposed to solve it. Its asymptotic and finite convergence are shown under certain assumptions. Numerical examples are presented to show how to solve DRO problems.

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arXiv2022

Dehomogenization for Completely Positive Tensors

A real symmetric tensor is completely positive (CP) if it is a sum of symmetric tensor powers of nonnegative vectors. We propose a dehomogenization approach for studying CP tensors. This gives new Moment-SOS relaxations for CP tensors. Detection for CP tensors and the linear conic optimization with CP tensor cones can be solved more efficiently by the dehomogenization approach.

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arXiv2022

Loss Functions for Finite Sets

This paper studies loss functions for finite sets. For a given finite set $S$, we give sum-of-square type loss functions of minimum degree. When $S$ is the vertex set of a standard simplex, we show such loss functions have no spurious minimizers (i.e., every local minimizer is a global one). Up to transformations, we give similar loss functions without spurious minimizers for general finite sets. When $S$ is approximately given by a sample set $T$, we show how to get loss functions by solving a quadratic optimization problem. Numerical experiments and applications are given to show the efficiency of these loss functions.

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arXiv2022

Homogenization for polynomial optimization with unbounded sets

This paper considers polynomial optimization with unbounded sets. We give a homogenization formulation and propose a hierarchy of Moment-SOS relaxations to solve it. Under the assumptions that the feasible set is closed at infinity and the ideal of homogenized equality constraining polynomials is real radical, we show that this hierarchy of Moment-SOS relaxations has finite convergence, if some optimality conditions (i.e., the linear independence constraint qualification, strict complementarity and second order sufficiency conditions) hold at every minimizer, including the one at infinity. Moreover, we prove extended versions of Putinar-Vasilescu type Positivstellensatz for polynomials that are nonnegative on unbounded sets. The classical Moment-SOS hierarchy with denominators is also studied. In particular, we give a positive answer to a conjecture of Mai, Lasserre and Magron in their recent work.

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arXiv2022

Low Rank Tensor Decompositions and Approximations

There exist linear relations among tensor entries of low rank tensors. These linear relations can be expressed by multi-linear polynomials, which are called generating polynomials. We use generating polynomials to compute tensor rank decompositions and low rank tensor approximations. We prove that this gives a quasi-optimal low rank tensor approximation if the given tensor is sufficiently close to a low rank one.

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arXiv2022

Generalized truncated moment problems with unbounded sets

This paper studies generalized truncated moment problems with unbounded sets. First, we study geometric properties of the truncated moment cone and its dual cone of nonnegative polynomials. By the technique of homogenization, we give a convergent hierarchy of Moment-SOS relaxations for approximating these cones. With them, we give a Moment-SOS method for solving generalized truncated moment problems with unbounded sets. Finitely atomic representing measures, or certificates for their nonexistence, can be obtained by the proposed method. Numerical experiments and applications are also given.

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arXiv2022

Higher Order Correlation Analysis for Multi-View Learning

Multi-view learning is frequently used in data science. The pairwise correlation maximization is a classical approach for exploring the consensus of multiple views. Since the pairwise correlation is inherent for two views, the extensions to more views can be diversified and the intrinsic interconnections among views are generally lost. To address this issue, we propose to maximize higher order correlations. This can be formulated as a low rank approximation problem with the higher order correlation tensor of multi-view data. We use the generating polynomial method to solve the low rank approximation problem. Numerical results on real multi-view data demonstrate that this method consistently outperforms prior existing methods.

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arXiv2021

Convex Generalized Nash Equilibrium Problems and Polynomial Optimization

This paper studies convex Generalized Nash Equilibrium Problems (GNEPs) that are given by polynomials. We use rational and parametric expressions for Lagrange multipliers to formulate efficient polynomial optimization for computing Generalized Nash Equilibria (GNEs). The Moment-SOS hierarchy of semidefinite relaxations are used to solve the polynomial optimization. Under some general assumptions, we prove the method can find a GNE if there exists one, or detect nonexistence of GNEs. Numerical experiments are presented to show the efficiency of the method.

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