Bilinear Problem Research Articles

A splitting preconditioned primal-dual algorithm with interpolation and extrapolation for bilinear saddle point problem

A splitting preconditioned primal-dual algorithm with interpolation and extrapolation for bilinear saddle point problem

On compressive self-calibration with structural constraints via alternating minimization

This paper considers the compressive bilinear self-calibration problem with explicit low-dimensional structural constraints. The celebrated proximal alternating linearized minimization (PALM) framework is adapted to simultaneously allow general sub-sampling schemes and structure-promoting regularizers. For the first time in literature, we refine the conditional convergence guarantees of PALM and show that the parameter commonly adopted to remove the scaling ambiguity as well as the structural penalties can ensure the unconditional convergence independent of strict assumptions on the statistical properties of the measurements, subspaces, number of snapshots, or initial iterates. In particular, we impose sparse and small total variation structures on the target signals and provide detailed numerical procedures for efficient computations. The extension to the complex-valued case is also made and extensive numerical experiments are carried out to corroborate the theoretical claims. Different choices of sub-sampling schemes and compression rates are simulated to support the effectiveness of the proposed algorithm under various settings. We also make comparisons with the state-of-art competing methods and the superiority of our proposed algorithm is empirically verified.

Distributionally Robust Portfolio Optimization under Marginal and Copula Ambiguity

AbstractWe investigate a new family of distributionally robust optimization problem under marginal and copula ambiguity with applications to portfolio optimization problems. The proposed model considers the ambiguity set of portfolio returns in which the marginal distributions and their copula are close—in terms of the Wasserstein distance—to their nominal counterparts. We develop a cutting-surface method to solve the proposed problem, in which the distribution separation subproblem is nonconvex and includes bilinear terms. We propose three approaches to solve the bilinear formulation, namely (1) linear relaxation via McCormick inequalities, (2) exact mixed-integer linear program reformulation via disjunctive inequalities, and (3) inner approximation method via a novel iterative procedure that exploits the structural properties of the bilinear optimization problem. We further carry out a comprehensive set of computational experiments with distributionally robust portfolios featuring Conditional Value-at-Risk (CVaR) measures. These tests aim to compare the accuracy of the proposed algorithms, analyze the impact of the radius of the Wasserstein ambiguity ball on the portfolio, and assess portfolio performance. We use a rolling-horizon approach to conduct the out-of-sample tests, which show the superior performance of the portfolios under marginal and copula ambiguity over the equally weighted and ambiguity-free Mean-CVaR benchmark portfolios.

A proactive/reactive mass screening approach with uncertain symptomatic cases.

We study the problem of mass screening of heterogeneous populations under limited testing budget. Mass screening is an essential tool that arises in various settings, e.g., the COVID-19 pandemic. The objective of mass screening is to classify the entire population as positive or negative for a disease as efficiently and accurately as possible. Under limited budget, testing facilities need to allocate a portion of the budget to target sub-populations (i.e., proactive screening) while reserving the remaining budget to screen for symptomatic cases (i.e., reactive screening). This paper addresses this decision problem by taking advantage of accessible population-level risk information to identify the optimal set of sub-populations for proactive/reactive screening. The framework also incorporates two widely used testing schemes: Individual and Dorfman group testing. By leveraging the special structure of the resulting bilinear optimization problem, we identify key structural properties, which in turn enable us to develop efficient solution schemes. Furthermore, we extend the model to accommodate customized testing schemes across different sub-populations and introduce a highly efficient heuristic solution algorithm for the generalized model. We conduct a comprehensive case study on COVID-19 in the US, utilizing geographically-based data. Numerical results demonstrate a significant improvement of up to 52% in total misclassifications compared to conventional screening strategies. In addition, our case study offers valuable managerial insights regarding the allocation of proactive/reactive measures and budget across diverse geographic regions.

Bilinear optimization of protein structure prediction: An exact approach via AB off-lattice model

Protein structure prediction (PSP) remains a central challenge in computational biology due to its inherent complexity and high dimensionality. While numerous heuristic approaches have appeared in the literature, their success varies. The AB off-lattice model, which characterizes proteins as sequences of A (hydrophobic) and B (hydrophilic) beads, presents a simplified perspective on PSP. This work presents a mathematical optimization-based methodology capitalizing on the off-lattice AB model. Dissecting the inherent non-linearities of the energy landscape of protein folding allowed for formulating the PSP as a bilinear optimization problem. This formulation was achieved by introducing auxiliary variables and constraints that encapsulate the nuanced relationship between the protein’s conformational space and its energy landscape. The proposed bilinear model exhibited notable accuracy in pinpointing the global minimum energy conformations on a benchmark dataset presented by the Protein Data Bank (PDB). Compared to traditional heuristic-based methods, this bilinear approach yielded exact solutions, reducing the likelihood of local minima entrapment. This research highlights the potential of reframing the traditionally non-linear protein structure prediction problem into a bilinear optimization problem through the off-lattice AB model. Such a transformation offers a route toward methodologies that can determine the global solution, challenging current PSP paradigms. Exploration into hybrid models, merging bilinear optimization and heuristic components, might present an avenue for balancing accuracy with computational efficiency.

Inverse optimization in semi-definite programs to impute unknown constraint matrices

In a typical (forward) optimization problem, a decision-maker uses given values of model parameters to compute the values of decision variables. The goal in inverse optimization (IO) is instead to infer parameters that render given values of decision variables optimal. Most papers on IO utilize duality to impute objective function parameters. A corresponding literature for imputing constraint parameters is essentially non-existent, even for linear programs. The difficulty is that these IO problems include nonconvex bilinear constraints and/or objectives. We study inverse semi-definite programs (SDPs) where matrices on the left-hand-sides of constraints are unknown. These unknown matrices must satisfy some side constraints. We consider two types of such problems, depending on which SDP decision variable values are given: primal or dual. In each situation, we study three sub-cases based on whether both the right-hand-side coefficients and the objective function matrix are known or only one of these two is known. This creates six variants of the inverse problem. For each variant, we first look for sufficient conditions, either on the side constraints or on problem structure, under which the nonconvex bilinear inverse problem can be simplified substantially. In particular, it either has a trivial solution or can be reformulated as a (set of) convex program(s). When such conditions are not evident, we propose three tailored heuristics called the Convex–Concave Procedure, Sequential Semi-definite Programming, and Alternate Convex Search. We perform computational experiments to gain insights into the performance of these three methods.

Identification of network interactions from time series data: An iterative approach.

The first step toward advancing our understanding of complex networks involves determining their connectivity structures from the time series data. These networks are often high-dimensional, and in practice, only a limited amount of data can be collected. In this work, we formulate the network inference task as a bilinear optimization problem and propose an iterative algorithm with sequential initialization to solve this bilinear program. We demonstrate the scalability of our approach to network size and its robustness against measurement noise, hyper-parameter variation, and deviations from the network model. Results across experimental and simulated datasets, comprising oscillatory, non-oscillatory, and chaotic dynamics, showcase the superior inference accuracy of our technique compared to existing methods.

Runtime Analysis of Competitive Co-evolutionary Algorithms for Maximin Optimisation of a Bilinear Function

Co-evolutionary algorithms have a wide range of applications, such as in hardware design, evolution of strategies for board games, and patching software bugs. However, these algorithms are poorly understood and applications are often limited by pathological behaviour, such as loss of gradient, relative over-generalisation, and mediocre objective stasis. It is an open challenge to develop a theory that can predict when co-evolutionary algorithms find solutions efficiently and reliable. This paper provides a first step in developing runtime analysis for population-based competitive co-evolutionary algorithms. We provide a mathematical framework for describing and reasoning about the performance of co-evolutionary processes. To illustrate the framework, we introduce a population-based co-evolutionary algorithm called PDCoEA, and prove that it obtains a solution to a bilinear maximin optimisation problem in expected polynomial time. Finally, we describe settings where PDCoEA needs exponential time with overwhelmingly high probability to obtain a solution.

The Meaning and Accuracy of the Improving Functions in the Solution of the CBQR by Krotov’s Method

A new solution to the continuous-time bilinear quadratic regulator optimal control problem (CBQR) was recently developed using Krotov’s Method. This paper provides two theoretical results related to the properties of that solution. The first discusses the equivalent representation of the cost-to-go performance index. The second one breaks down this equivalence into smaller identities referencing the components of the performance index. The paper shows how these results can be used to verify the numerical accuracy of the computed solution. Additionally, the meaning of the improving function and the equivalent representation, which are the main elements in the discussed CBQR’s solution, are explained according to the derived notions. A numerical example of structural control application exemplifies the significance of these results and how they can be applied to a specific CBQR problem.

Solving Sparse Separable Bilinear Programs Using Lifted Bilinear Cover Inequalities

Recently a class of second-order cone representable convex inequalities called lifted bilinear cover inequalities were introduced, which are valid for a set described by a separable bilinear constraint together with bounds on variables. In this paper, we study the computational potential of these inequalities for separable bilinear optimization problems. We first prove that the semidefinite programming relaxation provides no benefit over the McCormick relaxation for such problems. We then design a simple randomized separation heuristic for lifted bilinear cover inequalities. In our computational experiments, we separate many rounds of these inequalities starting from McCormick’s relaxation of instances where each constraint is a separable bilinear constraint set. We demonstrate that there is a significant improvement in the performance of a state-of-the-art global solver in terms of gap closed, when these inequalities are added at the root node compared with when they are not. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms – Discrete. Funding: S. S. Dey gratefully acknowledges the support by the Office of Naval Research [Grant N000141912323].

Level-set based topology optimization for bi-linear type elasto-plastic problems

Level-set based topology optimization for bi-linear type elasto-plastic problems

Error estimates for the discretization of bilinear control problems governed by semilinear elliptic PDEs

Error estimates for the discretization of bilinear control problems governed by semilinear elliptic PDEs

Control Problem Related to a 2D Parabolic–Elliptic Chemo-Repulsion System with Nonlinear Production

In this work, we analyze a bilinear optimal control problem related to a 2D parabolic–elliptic chemo-repulsion system with a nonlinear chemical signal production term. We prove the existence of global optimal solutions with bilinear control, and applying a generic result on the existence of Lagrange multipliers in Banach spaces, we obtain first-order necessary optimality conditions and derive an optimality system for a local optimal solution.

Metalearning-Based Alternating Minimization Algorithm for Nonconvex Optimization.

In this article, we propose a novel solution for nonconvex problems of multiple variables, especially for those typically solved by an alternating minimization (AM) strategy that splits the original optimization problem into a set of subproblems corresponding to each variable and then iteratively optimizes each subproblem using a fixed updating rule. However, due to the intrinsic nonconvexity of the original optimization problem, the optimization can be trapped into a spurious local minimum even when each subproblem can be optimally solved at each iteration. Meanwhile, learning-based approaches, such as deep unfolding algorithms, have gained popularity for nonconvex optimization; however, they are highly limited by the availability of labeled data and insufficient explainability. To tackle these issues, we propose a meta-learning based alternating minimization (MLAM) method that aims to minimize a part of the global losses over iterations instead of carrying minimization on each subproblem, and it tends to learn an adaptive strategy to replace the handcrafted counterpart resulting in advance on superior performance. The proposed MLAM maintains the original algorithmic principle, providing certain interpretability. We evaluate the proposed method on two representative problems, namely, bilinear inverse problem: matrix completion and nonlinear problem: Gaussian mixture models. The experimental results validate the proposed approach outperforms AM-based methods.

On the Linear Convergence of Extragradient Methods for Nonconvex–Nonconcave Minimax Problems

Recently, minimax optimization has received renewed focus due to modern applications in machine learning, robust optimization, and reinforcement learning. The scale of these applications naturally leads to the use of first-order methods. However, the nonconvexities and nonconcavities present in these problems, prevents the application of typical gradient descent/ascent, which is known to diverge even in bilinear problems. Recently, it was shown that the proximal point method (PPM) converges linearly for a family of nonconvex–nonconcave problems. In this paper, we study the convergence of a damped version of the extragradient method (EGM), which avoids potentially costly proximal computations, relying only on gradient evaluation. We show that the EGM converges linearly for smooth minimax optimization problems satisfying the same nonconvex–nonconcave condition needed by the PPM. Funding: H. Lu was supported by The University of Chicago Booth School of Business Benjamin Grimmer was supported by Johns Hopkins Applied Mathematics and Statistics Department.

Second-order time discretization for reaction coefficient estimation of bilinear parabolic optimization problem with Neumann boundary conditions

Second-order time discretization for reaction coefficient estimation of bilinear parabolic optimization problem with Neumann boundary conditions

Set-theoretic output feedback control: A bilinear programming approach

This paper addresses the problem of designing output feedback controllers for constrained linear systems subject to bounded process and measurement disturbances. The proposed solution extends the set-theoretic model predictive control framework to deal with constrained output regulation problems. In particular, this is here achieved by adequately exploiting the extended Farka’s lemma to offline compute through bilinear optimization problems, a family of robust one-step controllable sets, and associated output feedback control gains. It is then formally proved that such computations can be online leveraged to design a simple switching controller capable of ensuring, by construction, that the state-trajectory of the system is always uniformly ultimately bounded, in a finite number of steps, into a small robust control invariant region. Finally, the effectiveness and benefits of the proposed solution are verified through a numerical example and compared with three alternative schemes.

A mesh‐based partitioning algorithm for decreasing conservatism in solving bilinear matrix inequality problems

SummaryIn this paper, a mesh‐based partitioning algorithm (MBPA) to solve bilinear matrix inequality (BMI) problems is proposed, where the main idea is to divide the solution space of a non‐convex BMI problem into smaller parts in the form of Simplexes through meshing the space. In each simplex, only the concave part of BMI constraints is approximated by piecewise affine matrix (PAM), which eventually leads to a convex sub‐problem. After that, the feasible solution can be easily obtained in each simplex through the linear matrix inequality (LMI) optimization. Although the proposed MBPA exploits PAM approximation, it relaxes several restrictive conditions used in other methods, so, it can be used as an effective approach for solving BMI problems. Various numerical examples on systems from COMPleib library to design static output feedback controller are provided to validate the proposed MBPA. Simulation results reveal the satisfactory performance of MBPA in several examined problems and provide acceptable performance compared to other existing BMI solution algorithms. Comparing the percentage of superiority of MBPA with some existing prominent BMI algorithms in several control problems including spectral abscissa optimization problems, and optimization problems for the closed‐loop systems of COMPleib show that the proposed algorithm is more successful than the previous ones.

The bang-bang property in some parabolic bilinear optimal control problems via two-scale asymptotic expansions

We investigate the bang-bang property for fairly general classes of L∞−L1 constrained bilinear optimal control problems in the case of parabolic equations set in the one-dimensional torus. Such a study is motivated by several applications in applied mathematics, most importantly in the study of reaction-diffusion models. The main equation writes ∂tum−∂xx2um=mum+f(t,x,um), where m=m(x) is the control, which must satisfy some L∞ bounds (0⩽m⩽1 a.e.) and an L1 constraint (∫m=m0 is fixed), and where f is a non-linearity that must only satisfy that any solution of this equation is positive at any given time. The functionals we seek to optimise are rather general; they write J(m)=∬(0,T)×Tj1(t,x,um)+∫Tj2(x,um(T,⋅)). Roughly speaking we prove in this article that, if j1 and j2 are increasing, then any maximiser m⁎ of J is bang-bang in the sense that it writes m⁎=1E for some subset E of the torus. It should be noted that such a result rewrites as an existence property for a shape optimisation problem. Our proof relies on second order optimality conditions, combined with a fine study of two-scale asymptotic expansions. In the conclusion of this article, we offer several possible generalisations of our results to more involved situations (for instance for controls of the form mφ(um) or for some time-dependent controls), and we discuss the limits of our methods by explaining which difficulties may arise in other settings.

On the Design of Constrained PI-Like Output Feedback Tracking Controllers via Robust Positive Invariance and Bilinear Programming

In this paper, we design a novel PI-like output feedback tracking controller for linear systems subject to state and input constraints. The proposed solution exploits the internal model principle to design via robust positive invariance and the Extended Farkas’ Lemma, a constrained set-point tracking controller. The considered polyhedral setup results in a bilinear optimization design problem that can be solved offline, allowing to simultaneously determine the sets of admissible set-points and plant’s initial conditions. The proposed solution guarantees asymptotic offset-free set-point tracking, local stability, and constraints fulfillment. Simulation results show the effectiveness of the proposed design and contrast it with an alternative scheme.