Semidefinite Optimization Problems Research Articles

A semidefinite programming approach for solving fractional optimal control problems

This paper presents a numerical scheme for solving fractional optimal control. The fractional derivative in this problem is in the Riemann–Liouville sense. The proposed method, based upon the method of moments, converts the fractional optimal control problem to a semidefinite optimization problem; namely, the nonlinear optimal control problem is converted to a convex optimization problem. The Grunwald–Letnikov formula is also used as an approximation for fractional derivative. The solution of fractional optimal control problem is found by solving the semidefinite optimization problem. Finally, numerical examples are presented to show the performance of the method.

An Interior Point Algorithm for Solving Convex Quadratic Semidefinite Optimization Problems Using a New Kernel Function

An Interior Point Algorithm for Solving Convex Quadratic Semidefinite Optimization Problems Using a New Kernel Function

Power Allocation for OFDM Wireless Network Localization Under Expectation and Robustness Constraints

In location-aware wireless networks, mobile nodes (agents) can obtain their positions using range measurements to other nodes with known positions (anchors). Optimal subcarrier power allocation at the anchors reduces positioning error and improves network lifetime and throughput. We present an optimization framework for subcarrier power allocations in network localization with the imperfect knowledge of network parameters based on the fundamental statistical limits. Power allocations with expectation and robustness constraints are obtained using semidefinite optimization problems in non-iterative and iterative forms with both unicast and multicast transmissions. Results show that the allocations provide more accurate localization than non-robust designs under channel and agents positions uncertainty.

Kronecker Product Constraints with an Application to the Two-Trust-Region Subproblem

We consider semidefinite optimization problems that include constraints of the form $G(x)\succeq 0$ and $H(x)\succeq 0$, where the components of the symmetric matrices $G(\cdot)$ and $H(\cdot)$ are affine functions of $x\in\mathbb{R}^n$. In such a case we obtain a new constraint $K(x,X)\succeq 0$, where the components of $K(\cdot,\cdot)$ are affine functions of $x$ and $X$, and $X$ is an $n\times n$ matrix that is a relaxation of $xx^T$. The constraint $K(x,X)\succeq 0$ is based on the fact that $G(x)\otimes H(x)\succeq 0$, where $\otimes$ denotes the Kronecker product. This construction of a constraint based on the Kronecker product generalizes the construction of a reformation-linearization technique (RLT) constraint from two linear inequality constraints, and also the construction of a second-order cone--RLT constraint from one linear inequality constraint and a second-order cone constraint. We show how the Kronecker product constraint obtained from two second-order cone constraints can be efficiently used to computationally strengthen the semidefinite programming relaxation of the two-trust-region subproblem.

A Mehrotra predictor-corrector interior-point algorithm for semidefinite optimization

This paper proposes a second-order Mehrotra-type predictor-corrector feasible interior-point algorithm for semidefinite optimization problems. In each iteration, the algorithm computes the Newton search directions through a new form of combination of the predictor and corrector directions. Using the Ai-Zhang wide neighborhood for linear complementarity problems, it is shown that the complexity bound of the algorithm is $O(\sqrt{n}\log \varepsilon^{-1})$ for the Nesterov-Todd search direction and $O({n}\log \varepsilon^{-1})$ for the Helmberg-Kojima-Monteiro search directions.

On semidefinite least squares and minimal unsatisfiability

This paper provides new results on the application of semidefinite optimization to satisfiability by studying the connection between semidefinite optimization and minimal unsatisfiability. We use a semidefinite least squares problem to assign weights to the clauses of a propositional formula in conjunctive normal form. We then show that these weights are a measure of the necessity of each clause in rendering the formula unsatisfiable, the weight of a necessary clause is strictly greater than the weight of any unnecessary clause. In particular, we show the following results: first, if a formula is minimal unsatisfiable, then all of its clauses have the same weight; second, if a clause does not belong to any minimal unsatisfiable subformula, then its weight is zero. An additional contribution of this paper is a demonstration of how the infeasibility of a semidefinite optimization problem can be tested using a semidefinite least squares problem by extending an earlier result for linear optimization. The connection between the semidefinite least squares problem and Farkas’ Lemma for semidefinite optimization is also discussed.

Smooth strongly convex interpolation and exact worst-case performance of first-order methods

We show that the exact worst-case performance of fixed-step first-order methods for unconstrained optimization of smooth (possibly strongly) convex functions can be obtained by solving convex programs. Finding the worst-case performance of a black-box first-order method is formulated as an optimization problem over a set of smooth (strongly) convex functions and initial conditions. We develop closed-form necessary and sufficient conditions for smooth (strongly) convex interpolation, which provide a finite representation for those functions. This allows us to reformulate the worst-case performance estimation problem as an equivalent finite dimension-independent semidefinite optimization problem, whose exact solution can be recovered up to numerical precision. Optimal solutions to this performance estimation problem provide both worst-case performance bounds and explicit functions matching them, as our smooth (strongly) convex interpolation procedure is constructive. Our works build on those of Drori and Teboulle (Math Program 145(1---2):451---482, 2014) who introduced and solved relaxations of the performance estimation problem for smooth convex functions. We apply our approach to different fixed-step first-order methods with several performance criteria, including objective function accuracy and gradient norm. We conjecture several numerically supported worst-case bounds on the performance of the fixed-step gradient, fast gradient and optimized gradient methods, both in the smooth convex and the smooth strongly convex cases, and deduce tight estimates of the optimal step size for the gradient method.

On the upper Lipschitz property of the KKT mapping for nonlinear semidefinite optimization

In this note, we prove that the KKT mapping for nonlinear semidefinite optimization problem is upper Lipschitz continuous in graph at the KKT point, under the second-order sufficient optimality conditions and the strict Robinson constraint qualification.

A Constraint-Reduced Algorithm for Semidefinite Optimization Problems with Superlinear Convergence

Constraint reduction is an essential method because the computational cost of the interior point methods can be effectively saved. Park and O'Leary proposed a constraint-reduced predictor---corrector algorithm for semidefinite programming with polynomial global convergence, but they did not show its superlinear convergence. We first develop a constraint-reduced algorithm for semidefinite programming having both polynomial global and superlinear local convergences. The new algorithm repeats a corrector step to have an iterate tangentially approach a central path, by which superlinear convergence can be achieved. This study proves its convergence rate and shows its effective cost saving in numerical experiments.

A primal–dual interior-point method for semidefinite optimization based on a class of trigonometric barrier functions

A primal–dual interior-point method (IPM) based on a new class of proximity functions is proposed for solving Semidefinite Optimization (SDO) problems. The proposed functions are induced from the kernel functions with trigonometric barrier terms. We derive iteration complexity of large-update IPMs for SDO as O(nlognlognϵ). This improves the result obtained in Li and Zhang (2015) for linear optimization and matches to the bound for the so-called self-regular kernel functions.

Linearly Solvable Stochastic Control Lyapunov Functions

This paper presents a new method for synthesizing stochastic control Lyapunov functions for a class of nonlinear stochastic control systems. The technique relies on a transformation of the classical nonlinear Hamilton-Jacobi-Bellman partial differential equation to a linear partial differential equation for a class of problems with a particular constraint on the stochastic forcing. This linear partial differential equation can then be relaxed to a linear differential inclusion, allowing for relaxed solutions to be generated using sum of squares programming. The resulting relaxed solutions are in fact viscosity super/subsolutions, and by the maximum principle are pointwise upper and lower bounds to the underlying value function, even for coarse polynomial approximations. Furthermore, the pointwise upper bound is shown to be a stochastic control Lyapunov function, yielding a method for generating nonlinear controllers with pointwise bounded distance from the optimal cost when using the optimal controller. These approximate solutions may be computed with non-increasing error via a hierarchy of semidefinite optimization problems. Finally, this paper develops a-priori bounds on trajectory suboptimality when using these approximate value functions, as well as demonstrates that these methods, and bounds, can be applied to a more general class of nonlinear systems not obeying the constraint on stochastic forcing. Simulated examples illustrate the methodology.

Fidelity between one bipartite quantum state and another undergoing local unitary dynamics

The fidelity and local unitary transformation are two widely useful notions in quantum physics. We study two constrained optimization problems in terms of the maximal and minimal fidelity between two bipartite quantum states undergoing local unitary dynamics. The problems are related to the geometric measure of entanglement and the distillability problem. We show that the problems can be reduced to semi-definite programming optimization problems. We give close-form formulae of the fidelity when the two states are both pure states, or a pure product state and the Werner state. We explain from the point of view of local unitary actions that why the entanglement in Werner states is hard to accessible. For general mixed states, we give upper and lower bounds of the fidelity using tools such as affine fidelity, channels and relative entropy from information theory. We also investigate the power of local unitaries and quantification for the commutativity of quantum states, and the equivalence of the two optimization problems.

Semidefinite relaxations for partitioning, assignment and ordering problems

Semidefinite optimization is a strong tool in the study of NP-hard combinatorial optimization problems. On the one hand, semidefinite optimization problems are in principle solvable in polynomial time (with fixed precision), on the other hand, their modeling power allows to naturally handle quadratic constraints. Contrary to linear optimization with the efficiency of the Simplex method, the algorithmic treatment of semidefinite problems is much more subtle and also practically quite expensive. This survey-type article is meant as an introduction for a non-expert to this exciting area. The basic concepts are explained on a mostly intuitive level, and pointers to advanced topics are given. We provide a variety of semidefinite optimization models on a selection of graph optimization problems and give a flavour of their practical impact.

Community detection in sparse networks via Grothendieck’s inequality

We present a simple and flexible method to prove consistency of semidefinite optimization problems on random graphs. The method is based on Grothendieck’s inequality. Unlike the previous uses of this inequality that lead to constant relative accuracy, we achieve any given relative accuracy by leveraging randomness. We illustrate the method with the problem of community detection in sparse networks, those with bounded average degrees. We demonstrate that even in this regime, various simple and natural semidefinite programs can be used to recover the community structure up to an arbitrarily small fraction of misclassified vertices. The method is general; it can be applied to a variety of stochastic models of networks and semidefinite programs.

Convex Solution to a Joint Attitude and Spin-Rate Estimation Problem

This paper considers the problem of jointly estimating the attitude and spin rate of a spinning spacecraft. Psiaki has formulated a family of optimization problems that generalize the classical least-squares attitude-estimation problem, known as Wahba’s problem, to the case of a spinning spacecraft. In the special case where the rotation axis and the fundamental sampling period are fixed and known, but the spin rate is unknown (such as for nutation-damped spin-stabilized spacecraft), Psiaki’s problem can be reformulated exactly as a type of tractable convex optimization problem called a semidefinite optimization problem. This reformulation allows the problem to be globally solved using standard numerical routines for semidefinite optimization.

SOCP relaxation bounds for the optimal subset selection problem applied to robust linear regression

This paper deals with the problem of finding the globally optimal subset of h elements from a larger set of n elements in d space dimensions so as to minimize a quadratic criterion, with an special emphasis on applications to computing the Least Trimmed Squares Estimator (LTSE) for robust regression. The computation of the LTSE is a challenging subset selection problem involving a nonlinear program with continuous and binary variables, linked in a highly nonlinear fashion. The selection of a globally optimal subset using the branch and bound (BB) algorithm is limited to problems in very low dimension, typically d ≤ 5, as the complexity of the problem increases exponentially with d. We introduce a bold pruning strategy in the BB algorithm that results in a significant reduction in computing time, at the price of a negligeable accuracy lost. The novelty of our algorithm is that the bounds at nodes of the BB tree come from pseudo-convexifications derived using a linearization technique with approximate bounds for the nonlinear terms. The approximate bounds are computed solving an auxiliary semidefinite optimization problem. We show through a computational study that our algorithm performs well in a wide set of the most difficult instances of the LTSE problem.

An $$O\left(\sqrt{n}L\right)$$ O n L wide neighborhood interior-point algorithm for semidefinite optimization

In this paper, we propose a primal–dual interior-point method for semidefinite optimization problems. The algorithm is based on a new class of search directions and the Ai-Zhang’s wide neighborhood for monotone linear complementarity problems. The theoretical complexity of the new algorithm is calculated. It is investigated that the proposed algorithm has polynomial iteration complexity \(O(\sqrt{n}L)\) and coincides with the best known iteration bound for semidefinite optimization problems.

Robust observation strategy to estimate the substrate concentration in the influent of a fermentative bioreactor for hydrogen production

The observation strategy developed in this paper consists of an H∞ Luenberger observer coupled to a classical super-twisting observer. The H∞ Luenberger observer uses the biohydrogen flow rate measured at the reactor output to estimate the glucose and the biomass concentrations inside the reactor. These estimations are taken by the super-twisting observer to estimate the glucose concentration at the reactor input. First, pseudo-stoichiometric and kinetic parameters of the bioreactor model are identified. Two semi-definite optimization problems are then proposed to compute the observer gains. Results show that the biomass is a sensitive state variable. In spite of this, the glucose estimated at the reactor input correctly follows the experimental data and the estimation error remains close to zero in the complete period of time considered.

A Polynomial Time Constraint-Reduced Algorithm for Semidefinite Optimization Problems

We present an infeasible primal-dual interior point method for semidefinite optimization problems, making use of constraint reduction. We show that the algorithm is globally convergent and has polynomial complexity, the first such complexity result for primal-dual constraint reduction algorithms for any class of problems. Our algorithm is a modification of one with no constraint reduction due to Potra and Sheng (1998) and can be applied whenever the data matrices are block diagonal. It thus solves as special cases any optimization problem that is a linear, convex quadratic, convex quadratically constrained, or second-order cone problem.

Complexity analysis of an interior point algorithm for the semidefinite optimization based on a kernel function with a double barrier term

In this paper, we establish the polynomial complexity of a primal-dual path-following interior point algorithm for solving semidefinite optimization (SDO) problems. The proposed algorithm is based on a new kernel function which differs from the existing kernel functions in which it has a double barrier term. With this function we define a new search direction and also a new proximity function for analyzing its complexity. We show that if q1 > q2 > 1, the algorithm has \(O((q_1 + 1)n^{\frac{{q_1 + 1}} {{2(q_1 - q_2 )}}} \log \tfrac{n} {e}) \) and \(O((q_1 + 1)^{\frac{{3q_1 - 2q_2 + 1}} {{2(q_1 - q_2 )}}} \sqrt n \log \tfrac{n} {e}) \) complexity results for large- and small-update methods, respectively.