mathematical-optimization.bigb

= Mathematical optimization
{wiki=Mathematical_optimization}

Mathematical optimization studies extrema under constraints.

= Lagrange sufficiency theorem
{parent=Mathematical optimization}

For a convex differentiable objective with convex differentiable inequality constraints, a feasible point satisfying the <Karush-Kuhn-Tucker conditions> with nonnegative multipliers is a global minimizer.

= Dynamic programming
{parent=Mathematical optimization}
{wiki}

Dynamic programming solves a multistage optimization problem backwards by expressing each remaining-horizon value in terms of the next-stage value.

= Bellman equation
{parent=Dynamic programming}
{c}
{wiki}

A Bellman equation is the recursive optimality relation for the <value function> of a <dynamic programming> problem.

= Value function
{parent=Dynamic programming}
{wiki=Value_function}

A value function assigns each state and time the best objective attainable from that state over the remaining decisions.

= Newton method
{parent=Mathematical optimization}
{c}
{wiki=Newton%27s_method_in_optimization}

Newton's method for minimizing a twice differentiable function uses
$$
x_{k+1}=x_k-[\nabla^2f(x_k)]^{-1}\nabla f(x_k).
$$

= Quadratic convergence bound for Newton's method
{parent=Newton method}

If $\nabla^2f\succeq mI$ and the Hessian is $M$-Lipschitz near a minimizer $x^*$, then
$$
\|x_{k+1}-x^*\|\leq\frac{M}{2m}\|x_k-x^*\|^2.
$$
Thus a sufficiently close initial point has an error exponent that doubles at each iteration.

= Karush-Kuhn-Tucker conditions
{c}
{parent=Mathematical optimization}
{wiki}

For differentiable inequality constraints $g_i(x)\leq0$, a regular local minimum has multipliers $\lambda_i\geq0$ satisfying
$$
\nabla f+\sum_i\lambda_i\nabla g_i=0,
\qquad
\lambda_i g_i=0.
$$
For a convex problem under a suitable constraint qualification, these conditions are sufficient for global optimality.

= Active-set transition in capped resource allocation
{parent=Karush-Kuhn-Tucker conditions}

In a convex resource-allocation problem, tightening an upper bound can change its multiplier from zero to positive. The optimum then moves from a point where only the total-resource constraint is active to the intersection where both constraints are active.

= Convex optimization
{parent=Mathematical optimization}
{wiki}

Convex optimization minimizes a convex objective over a convex feasible set.

= Subdifferential
{title2=$\partial f(x)$}
{parent=Convex optimization}
{wiki}

For a convex function $f$, the subdifferential at $x$ is
$$
\partial f(x)=\{g:f(y)\geq f(x)+g^T(y-x)\text{ for every }y\}.
$$
The point $x$ minimizes $f$ exactly when $0\in\partial f(x)$.

= Stiemke theorem
{parent=Convex optimization}
{c}
{wiki=Gordan%27s_theorem}

For a real matrix $P$, exactly one of the following holds: there is $\phi$ with $P\phi\geq0$ and $P\phi\ne0$, or there is $q>0$ with $P^Tq=0$. Separating $\operatorname{Im}P$ from the standard simplex proves the result; $q$ may then be normalized to sum to one.

= Projected gradient descent
{parent=Convex optimization}
{wiki=Proximal_gradient_methods_for_learning}

For a convex set $C$ and a differentiable convex objective $F$, projected gradient descent uses
$$
x_{i+1}=\Pi_C(x_i-\eta\nabla F(x_i)).
$$

= Averaged projected-gradient bound
{parent=Projected gradient descent}

If $\|\nabla F(x_i)\|\leq G$ and $\|x_1-x_*\|\leq D$, then
$$
F\left(\frac1k\sum_{i=1}^kx_i\right)-F(x_*)
\leq\frac{D^2}{2\eta k}+\frac{\eta G^2}{2}.
$$
It follows by expanding the squared distance after each projected step, using nonexpansiveness of projection, summing the resulting inequalities, and applying convexity.

= Lagrange multiplier
{parent=Mathematical optimization}
{c}
{wiki}

At a regular constrained extremum of $f$ subject to $g=0$, the gradients satisfy
$$
\nabla f=\lambda\nabla g.
$$

= Derivative of a constrained value function
{parent=Lagrange multiplier}

For
$$
\phi(b)=\inf\{f(x):g(x)=b\},
$$
suppose the optimizer and multiplier vary smoothly and use $L=f-\lambda(g-b)$. Differentiating the optimum value and using stationarity gives
$$
\phi'(b)=\lambda(b).
$$

= Weighted open-box minimization
{parent=Lagrange multiplier}

For a topless rectangular box with dimensions $x,y,z$, volume $V$, and weighted face cost
$$
A=axy+bxz+cyz,
$$
an interior optimum equalizes the three terms:
$$
axy=bxz=cyz.
$$
The same result and global minimality follow from the arithmetic-geometric mean inequality.

= Arithmetic-geometric mean inequality
{parent=Mathematical optimization}
{wiki}

For nonnegative $a_1,\ldots,a_n$,
$$
\frac{a_1+\cdots+a_n}{n}\geq(a_1\cdots a_n)^{1/n},
$$
with equality exactly when all the $a_i$ are equal.

= Linear programming
{parent=Mathematical optimization}
{wiki}

Linear programming optimizes a <linear function> subject to finitely many linear equalities and inequalities.

= Basic feasible solution
{parent=Linear programming}
{wiki}

For $Ax=b$, $x\geq0$, a feasible vector $x$ is basic when the columns $A_i$ indexed by its positive coordinates are linearly independent. Equivalently, enlarge those columns to a basis, set every nonbasic coordinate to zero, and solve the resulting square system.

= Fundamental theorem of linear programming
{parent=Basic feasible solution}
{c}
{wiki=Fundamental_theorem_of_linear_programming}

If a linear program has an optimal solution, it has an optimal <basic feasible solution>. Choose an optimum with minimal positive support. A dependence among its active columns gives a feasible two-sided perturbation; optimality makes its objective slope zero, and moving until one coordinate vanishes contradicts minimality.

= Linear-fractional programming
{parent=Linear programming}
{wiki}

Linear-fractional programming optimizes a ratio $c^Tx/d^Tx$ over a polyhedron on which the denominator is positive.

= Charnes-Cooper transformation
{parent=Linear-fractional programming}
{c}
{wiki}

For $d^Tx>0$, set
$$
y=\frac{x}{d^Tx},
\qquad
t=\frac1{d^Tx}.
$$
Then $d^Ty=1$, and $Ax=b$ becomes $Ay=bt$, turning a linear-fractional program into a linear program with objective $c^Ty$.

= Linear programming duality
{parent=Linear programming}
{wiki=Dual_linear_program}

Every linear maximization program has a dual minimization program whose feasible objective values bound the primal values.

= Dual linear program
{parent=Linear programming duality}

For a primal maximization problem $Ax\leq b$, $x\geq0$, the dual minimizes $b^Ty$ subject to $A^Ty\geq c$, $y\geq0$.

= Dual of a minimization linear program in inequality form
{parent=Linear programming duality}

The dual pair
$$
\min\{c^Tx:Ax\geq b,\ x\geq0\}
\quad\hbox{and}\quad
\max\{b^Ty:A^Ty\leq c,\ y\geq0\}
$$
satisfies weak duality because $b^Ty\leq y^TAx=x^TA^Ty\leq c^Tx$ for every feasible pair.

= Weak duality
{parent=Linear programming duality}

Every feasible dual objective value bounds every feasible primal objective value.

= Complementary slackness
{parent=Linear programming duality}
{wiki}

At a primal-dual optimum, each positive variable corresponds to a tight dual constraint and conversely for positive dual variables.

= Transportation problem
{parent=Linear programming duality}
{wiki=Transportation_theory_(mathematics)}

The transportation problem minimizes linear shipping cost while matching prescribed row supplies and column demands.

= Transportation polytope
{parent=Transportation problem}
{wiki}

The transportation polytope is the set of nonnegative matrices with fixed row and column sums.

= Transportation spanning tree
{parent=Transportation polytope}

The positive cells of a nondegenerate transportation basis form a spanning tree in the bipartite graph of supply and demand vertices.

= Transportation simplex algorithm
{parent=Transportation problem}
{wiki=Transportation_problem\#Solution}

The transportation simplex computes row and column potentials, enters a negative-reduced-cost cell, and pivots around its induced alternating cycle.

= Northwest corner method
{parent=Transportation simplex algorithm}
{wiki}

The northwest-corner method repeatedly allocates the smaller remaining supply and demand in the current top-left cell to construct an initial transportation basis.

= Reduced cost
{parent=Transportation simplex algorithm}
{wiki}

For transportation potentials, the reduced cost is $\bar c_{ij}=c_{ij}-u_i-v_j$; a negative value identifies an improving entering cell.

= Cycle pivot
{parent=Transportation simplex algorithm}

Adding a nonbasic edge to a transportation tree creates one cycle; alternating equal additions and subtractions around it preserves every row and column sum.

= Integrality of the transportation problem
{parent=Transportation problem}

Integer supplies and demands admit an integer optimum because an integer initial basis and every cycle pivot remain integer.

= Totally unimodular matrix
{parent=Integrality of the transportation problem}
{wiki}

A matrix is totally unimodular when every square subdeterminant is $0$, $1$, or $-1$; linear programs with such a constraint matrix and integer right side have integral vertices.

= Simplex method
{parent=Linear programming}
{wiki}

The simplex method moves between adjacent <basic feasible solutions> along improving edges of a feasible polytope until no improving pivot remains.

= Simplex paths on a cube with one truncated corner
{parent=Simplex method}

For $0\leq x_i\leq1$, $x_1+x_2+x_3\leq5/2$, and objective $x_1+2x_2+4x_3$, the oriented edge graph has eight improving paths from the origin to the unique optimum $(1/2,1,1)$. Their lengths range from three to five pivots.

= Convex set
{parent=Mathematical optimization}
{wiki}

A convex set contains every <line segment> joining two of its points.

= Nonnegative orthant
{title2=$\mathbb R_{\geq0}^n$}
{parent=Convex set}
{wiki=Orthant}

The nonnegative orthant consists of vectors whose coordinates are all nonnegative. Its interior is the positive orthant, where every coordinate is positive.

= Convex combination
{parent=Convex set}
{wiki}

A convex combination of points $x_1,\ldots,x_n$ is a sum
$$
\sum_{i=1}^n\lambda_i x_i,
\qquad \lambda_i\geq0,
\qquad \sum_{i=1}^n\lambda_i=1.
$$

= Cyclic symmetry averaging
{parent=Convex combination}

If a convex subset of $\mathbb R^n$ is invariant under cyclic coordinate permutation, averaging the $n$ cyclic images of $x$ shows that
$$
\left(\frac1n\sum_jx_j,\ldots,\frac1n\sum_jx_j\right)
$$
also belongs to the set.

= Closed half-space
{parent=Convex set}
{wiki=Half-space_(geometry)}

A closed half-space is a set of the form $\{x:a^Tx\leq b\}$ for a nonzero vector $a$; it is a <convex set> and is closed.

= Line segment
{parent=Convex set}
{wiki}

The line segment joining vectors $x$ and $y$ is
$$
\{(1-t)x+ty:0\leq t\leq1\}.
$$

= Convex polytope
{parent=Convex set}
{wiki=Polytope}

A convex polytope is a bounded intersection of finitely many closed half-spaces, equivalently the convex hull of finitely many points.

= Extreme point
{parent=Convex set}
{wiki}

An extreme point of a convex set $C$ is a point $x\in C$ for which
$$
x=(1-t)y+tz,\qquad y,z\in C,\quad0<t<1
$$
implies $x=y=z$.

= Euclidean projection onto a convex set
{parent=Convex set}
{wiki=Projection_onto_convex_sets}

For a nonempty closed convex subset $C$ of a finite-dimensional inner-product space, $\Pi_C(x)$ is the unique point of $C$ minimizing $\|x-z\|$.

= Variational characterization of convex projection
{parent=Euclidean projection onto a convex set}

For $p\in C$,
$$
p=\Pi_C(x)
\quad\Longleftrightarrow\quad
\langle x-p,z-p\rangle\leq0
\quad\text{for every }z\in C.
$$

= Second-order cone
{parent=Convex set}
{wiki=Second-order_cone_programming}

The second-order cone, also called the Lorentz cone, is
$$
\mathcal L_{p+1}=\{(v,s)\in\mathbb R^p\times\mathbb R:\|v\|_2\leq s\}.
$$

= Lorentz cone
{synonym}

= Projection onto the second-order cone
{parent=Second-order cone}

For $x=(u,t)$ and $\rho=\|u\|_2$,
$$
\Pi_{\mathcal L_{p+1}}(u,t)=
\begin{cases}
(u,t),&\rho\leq t,\\
(0,0),&\rho\leq-t,\\
\displaystyle\frac12\left(1+\frac t\rho\right)(u,\rho),&\rho>|t|.
\end{cases}
$$

= Convex hull
{parent=Convex set}
{wiki}

The convex hull of a set is the set of all finite convex combinations of its points, equivalently the smallest convex set containing it.

= Caratheodory theorem
{c}
{parent=Convex hull}
{wiki=Carath%C3%A9odory%27s_theorem_(convex_hull)}

Every point in the convex hull of a subset of $\mathbb R^d$ is a convex combination of at most $d+1$ points of that subset.

= Positive semidefinite cone
{parent=Convex set}
{wiki=Definite_matrix}

The positive semidefinite matrices form a convex cone because nonnegative combinations preserve nonnegative quadratic forms.

= Trace constraint
{parent=Positive semidefinite cone}

For a positive semidefinite matrix, a trace bound is a bound on the sum of its nonnegative eigenvalues.

= Positive semidefinite trace ball
{parent=Trace constraint}

The positive semidefinite trace ball of radius $s$ is
$$
\mathcal S_s=\{Z=Z^T:Z\succeq0,\ \operatorname{tr}Z\leq s\}.
$$
It is closed and convex, and $\|Z\|_F\leq\operatorname{tr}Z\leq s$ for every $Z\in\mathcal S_s$.

= Projection onto a positive semidefinite trace ball
{parent=Positive semidefinite trace ball}

If $M\succeq0$ has eigenpairs $(\mu_i,v_i)$ and $\operatorname{tr}M>s$, then its Frobenius projection onto $\mathcal S_s$ is
$$
\Pi_{\mathcal S_s}(M)
=\sum_i(\mu_i-\rho)_+v_iv_i^T,
$$
where $\rho>0$ is uniquely determined by $\sum_i(\mu_i-\rho)_+=s$.

= Game theory
{parent=Mathematical optimization}
{wiki}

Game theory studies strategic interaction among decision makers.

= Mixed strategy
{parent=Game theory}
{wiki}

A mixed strategy is a probability distribution over a player's pure strategies.

= Bimatrix game
{parent=Game theory}
{wiki}

A bimatrix game is a finite two-player game specified by one payoff matrix for each player.

= Nash equilibrium
{c}
{parent=Bimatrix game}
{wiki}

A Nash equilibrium is a strategy profile in which no player can increase their expected payoff by changing only their own strategy.

= Brouwer proof of Nash equilibrium for a two-by-two game
{parent=Nash equilibrium}

Normalize each player's positive pure-strategy payoff improvements to define a continuous self-map of the product of their mixed-strategy simplices. At a Brouwer fixed point every positive improvement must vanish, giving a Nash equilibrium.

= Zero-sum game
{parent=Game theory}
{wiki}

In a two-player zero-sum game, one player’s payoff is the other player’s loss.

= Matrix game
{parent=Zero-sum game}
{wiki=Zero-sum_game\#Solution}

A matrix game is a finite two-player zero-sum game whose entry $A_{ij}$ is the row player's payoff when the players choose row $i$ and column $j$.

= Matrix-game optimization problem
{parent=Matrix game}

The row player solves
$$
\max_{p,v}v
\quad\text{subject to}\quad
A^Tp\geq ve,quad e^Tp=1,quad p\geq0.
$$

= Mixed-strategy optimality certificate for a matrix game
{parent=Matrix game}

If the column player has an optimal strategy at value $v$ and a row probability vector $p$ satisfies $p^TA\geq ve^T$, then $p$ is optimal: it guarantees $v$, while the optimal column strategy prevents every row strategy from exceeding $v$.

= Symmetric inverse formula for a matrix-game equilibrium
{parent=Matrix game}

If a symmetric invertible payoff matrix satisfies $A^{-1}e\geq0$, both players may use
$$
p=q=\frac{A^{-1}e}{e^TA^{-1}e},
$$
and the game value is $(e^TA^{-1}e)^{-1}$.

= Three-card threshold-sum zero-sum game
{parent=Symmetric inverse formula for a matrix-game equilibrium}

For
$$
A=\begin{pmatrix}2&3&4\\3&4&-5\\4&-5&-6\end{pmatrix},
$$
both players optimally choose cards $1,2,3$ with probabilities $41/50,2/25,1/10$, and the row player's expected payoff is $57/25$.

= Minimax theorem
{parent=Zero-sum game}
{wiki}

The finite minimax theorem equates the maximizing player’s security level with the minimizing player’s security level.

= Optimal mixed strategy
{parent=Zero-sum game}
{wiki}

An optimal mixed strategy guarantees the value of the game against every opposing pure strategy.

= Antisymmetric zero-sum game
{parent=Zero-sum game}
{wiki}

A square game with antisymmetric payoff matrix has value zero.

= Dominated strategy
{parent=Game theory}
{wiki}

A dominated strategy can be replaced by another strategy that performs at least as well against every opponent action.

= Mathematical finance
{parent=Mathematical optimization}
{wiki=Mathematical_finance}

Mathematical finance applies probability and optimization to portfolios, prices, and risk.

= Arbitrage
{parent=Mathematical finance}
{wiki=Arbitrage}

An arbitrage is a zero-cost <self-financing portfolio> whose terminal payoff is nonnegative in every state and strictly positive with positive probability.

= Self-financing portfolio
{parent=Arbitrage}
{wiki}

A portfolio is self-financing when every change in its asset holdings is paid for entirely by selling or buying assets within the portfolio, with no external cash added or removed.

= Utility function
{parent=Mathematical finance}
{wiki=Utility}

In expected-utility portfolio choice, an increasing utility function prefers greater wealth, while concavity models aversion to risk.

= Expected utility maximization
{parent=Utility function}
{wiki=Expected_utility_hypothesis}

Expected utility maximization chooses an admissible random payoff $X$ to maximize $\mathbb E[U(X)]$.

= Utility indifference price
{parent=Expected utility maximization}
{wiki=Indifference_price}

A utility indifference price for a claim is the cash amount at which optimally buying the claim leaves the investor with the same maximal expected utility as trading without it.

= Optimized affine shift of concave utility
{parent=Expected utility maximization}

Let $\mathcal X$ be a vector space of random variables and suppose
$$
F(y)=\sup_{X\in\mathcal X}\mathbb E[U(X+y)]
$$
has an optimizer for every $y$. If $U$ is increasing and concave, then $F$ is increasing and concave. For concavity, combine optimizers at two values using the same convex coefficient as the values themselves.

= Scaled centered risk under concave utility
{parent=Expected utility maximization}

If $U$ is concave and $\mathbb E Z=0$, then
$$
G(s)=\mathbb E[U(m+sZ)]
$$
is concave in $s$. By <Jensen inequality>, $G(s)\leq G(0)$; concavity then makes $G$ nonincreasing on $[0,\infty)$.

= Discrete-time expected-utility portfolio problem
{parent=Mathematical finance}

For independent return innovations $\xi_n$ and a <predictable process> of holdings $\theta_n$, wealth obeys
$$
X_n=(1+r)X_{n-1}+\theta_n^T\xi_n.
$$
The investor chooses the holdings to maximize expected utility of terminal wealth.

= Bellman equation for terminal-wealth utility
{parent=Discrete-time expected-utility portfolio problem}

For terminal utility $U$ and independent return innovations,
$$
V(N,x)=U(x),\qquad
V(n,x)=\sup_{\theta\in\mathbb R^d}
\mathbb E\left[V\left(n+1,(1+r)x+\theta^T\xi_{n+1}\right)\right].
$$

= Monotonicity and concavity of a portfolio value function
{parent=Bellman equation for terminal-wealth utility}

If terminal utility is increasing and concave and each Bellman supremum is attained, backward induction shows that every remaining-horizon value function is increasing and concave. Concavity follows by combining optimal portfolios for two initial wealths with the same convex coefficient.

= Equivalent martingale measure
{parent=Mathematical finance}
{wiki=Equivalent_martingale_measure}

An equivalent martingale measure is a probability measure equivalent to the physical measure under which discounted asset prices are martingales. In a finite one-period market, its existence is equivalent to no arbitrage.

= Discrete-time binomial market
{parent=Mathematical finance}
{wiki=Binomial_options_pricing_model}

At each period a risky asset in a binomial market has one of two returns, while the risk-free asset grows by a fixed factor. When the risk-free return lies strictly between the two stock returns, the market is arbitrage-free and complete.

= Risk-neutral probability in a binomial market
{parent=Discrete-time binomial market}

If the stock factors are $1+a$ and $1+b$ and the risk-free factor is $1+r$, the discounted stock is a martingale under the probability
$$
q=\frac{r-a}{b-a}
$$
of an up move. The probability of a down move is $(b-r)/(b-a)$.

= Replicating portfolio in a binomial market
{parent=Discrete-time binomial market}

For one-period successor claim values $V_u,V_d$ and stock prices $S_u,S_d$, the replicating stock holding is
$$
\Delta=\frac{V_u-V_d}{S_u-S_d}.
$$
The remaining value is placed in the risk-free asset. In a complete binomial market, backward replication determines the unique no-arbitrage claim price.

= Backward option pricing
{parent=Replicating portfolio in a binomial market}

At each node of a binomial tree, a claim with next-period values $V_u,V_d$ has value
$$
V=\frac{qV_u+(1-q)V_d}{1+r},
$$
where $q$ is the local <risk-neutral probability in a binomial market>. For an American claim, replace this continuation value by the maximum of continuation and immediate exercise value.

= Stock-numeraire measure in a binomial market
{parent=Discrete-time binomial market}

Let $Q$ be the risk-neutral measure in an $N$-period binomial market. The stock-numeraire measure is defined on terminal events by
$$
\widehat Q(A)=\frac{\mathbb E_Q[S_N\mathbf1_A]}{S_0(1+r)^N}.
$$
If the $Q$-probability of an up move is $q$, its up probability under $\widehat Q$ is
$$
\widehat q=\frac{q(1+b)}{1+r}.
$$
This change of measure converts discounted expectations containing a factor $S_N$ into probabilities under $\widehat Q$.

= Fundamental theorem of asset pricing
{parent=Mathematical finance}
{wiki}

In a finite market, absence of arbitrage is equivalent to the existence of an equivalent martingale measure. If that measure is unique, every contingent claim has a unique no-arbitrage price given by its discounted expectation.

= Finite-state superhedging alternative
{parent=Fundamental theorem of asset pricing}

Let $P$ be the state-by-asset matrix of discounted gains. There exists $\theta$ with $Y-P\theta\geq0$ exactly when
$$
q^TY\geq0
$$
for every $q\geq0$ satisfying $P^Tq=0$. This is the finite-dimensional cone-separation, or Farkas-duality, form of superhedging.

= European call option
{parent=Fundamental theorem of asset pricing}
{wiki}

A European call with maturity $N$ and strike $K$ pays $(S_N-K)^+$ at time $N$.

= Put-call parity
{parent=European call option}
{wiki}

For European calls and puts with the same maturity and strike,
$$
C_0-P_0=S_0-(1+r)^{-N}K.
$$

= Binomial call-price recursion
{parent=European call option}

Conditioning on the first binomial move expresses a call with $N+1$ periods as a positive weighted sum of two $N$-period calls with rescaled strikes.

= Forward-start call option
{parent=European call option}
{wiki}

A forward-start call fixes its strike at a future time as a multiple of the stock price then. In a homogeneous binomial model, its time-zero price reduces to that of an ordinary call over the remaining periods.

= European put option
{parent=Fundamental theorem of asset pricing}
{wiki=Put_option}

A European put with maturity $N$ and strike $K$ pays $(K-S_N)^+$ at time $N$ and can be exercised only at maturity.

= American option
{parent=Mathematical finance}
{wiki}

An American option may be exercised at any time up to maturity. Its value is the <Snell envelope> of its discounted exercise payoff.

= American put option
{parent=American option}
{wiki=Put_option}

An American put has exercise payoff $(K-S_n)^+$. In a finite binomial market, <backward option pricing> compares that payoff with the discounted risk-neutral continuation value at every node.

= Mean-variance optimization
{parent=Mathematical finance}
{wiki=Modern_portfolio_theory}

Mean-variance optimization balances expected linear return against a positive-definite quadratic variance penalty.

= Mean-variance efficient ray
{parent=Mean-variance optimization}

Without a risk-free intercept or constraints, every undominated linear portfolio lies on the nonnegative ray through $V^{-1}b$.

= Gaussian one-fund theorem
{parent=Mean-variance optimization}
{c}

Let $\xi\sim N(b,\Sigma)$, where $b\ne0$ and $\Sigma$ is positive definite. For any increasing concave objective of $m+\theta^T\xi$, every unique optimal portfolio has the form
$$
\theta^*=\lambda\Sigma^{-1}b,
\qquad \lambda\geq0.
$$
Indeed, the component orthogonal to $\Sigma^{-1}b$ in the $\Sigma$ inner product contributes independent mean-zero Gaussian risk without changing the mean. Removing it cannot reduce expected concave utility. A negative coefficient is dominated by the corresponding positive coefficient, which has the same variance and a larger mean.

= Pareto dominance in mean-variance space
{parent=Mean-variance optimization}
{c}

A portfolio dominates another when it has no smaller mean and no larger variance, with at least one strict improvement.

= Mean-variance portfolio regression
{parent=Mean-variance optimization}

Projection onto the mean-variance portfolio $V^{-1}b$ leaves residual covariance $V-bb^T/(b^TV^{-1}b)$.

= Positive-definite quadratic optimization
{parent=Mathematical optimization}

A linear functional minus one half of a positive-definite quadratic has the unique maximizer obtained by solving its linear first-order equation.

= Orthogonal decomposition in a positive-definite metric
{parent=Positive-definite quadratic optimization}

A positive-definite matrix $V$ defines the inner product $x^TVy$, allowing decomposition into a chosen direction and its $V$-orthogonal complement.