Sen. Thad Cochran (R-Miss.) waves to supporters as he leaves a stop on the first day of a three-week campaign Wednesday. (Rogelio V. Solis/AP)

Sen. John Cornyn (R-Tex.) chose a military metaphor to his approach: “My strategy was the same as Norman Schwarzkopf’s in Desert Storm: overwhelming force. We didn’t want to have a fair fight.”

For Sen. Lamar Alexander (R-Tenn.), the key is a constant presence all over his state. “The most important thing to do is stay connected to the voters,” Alexander said.

After five long years of experimentation, veteran Senate Republicans now say there are a few clear paths for easily beating back a tea party challenge. Sen. Thad Cochran (R-Miss.) took a separate route and now finds himself in a desperate three-week sprint against a GOP state senator, hoping that the Republican establishment will prop him up against an onslaught of money from Washington-based conservative groups.

Cochran was thrown into a June 24 runoff against Chris McDaniel after both men fell short of the 50 percent threshold needed to win a primary in Mississippi. If Cochran loses, it will happen at least in part because of a campaign that bears all the hallmarks of a veteran who had no feel for modern political combat or a full grasp of insurgent challenges to the GOP establishment. He would be the fourth Senate Republican to lose in a primary since 2010.

Publicly and privately, Republicans are bemoaning the incumbent’s campaign operation, from the candidate’s own stumbles on the stump to his lack of preparation for the race. Cochran would be the first incumbent senator to fall in the 2014 primary season, and, despite conservative activists fielding an unprecedented number of challengers, no other GOP incumbent is imperiled.

As Cornyn, Alexander and other Republicans demonstrated this spring, it did not have to be this way for Cochran.

Both first elected in 2002, Cornyn and Alexander seemed ripe for a strong tea-party challenge. Cornyn and Alexander have served various stints in leadership and helped broker their share of bipartisan deals. On the hot-button issue of immigration, Cornyn helped craft a bipartisan plan that fell apart in 2007, and last year Alexander joined a group of Republicans who voted for the Senate’s comprehensive overhaul of the immigration system.

Yet Cornyn crushed a collection of Republican opponents in March and, two months ahead of his primary, Alexander is cruising to renomination.

It’s a similar circumstance for Senate Minority Leader Mitch McConnell (R-Ky.), who won his primary by more than 25 percentage points, and Sen. Lindsey O. Graham (R.-S.C.), who is well ahead in the latest polling before Tuesday’s primary.

Each of those senators applied a version of Cornyn’s “overwhelming force” doctrine.

Cornyn, for instance, was sitting on more than $6.9 million last Oct. 1, according to federal reports, two months before Texas candidates had to file for office. Graham socked away $7.6 million in his campaign account by Jan. 1, less than three months before the South Carolina filing deadline.

They had effectively fired warning shots to any potential challenger, as well as outside groups such as the Club for Growth and Senate Conservatives Fund. It would take a massive amount of cash to seriously challenge Cornyn and Graham, and the conservative groups took a pass on both races.

Still, in Texas, Rep. Steve Stockman — a colorful conservative with questionable personal finances and little fundraising history — filed to run against Cornyn, as did a few other low-profile Republicans. In South Carolina, a collection of local politicians with little visibility entered the GOP race against Graham.

Despite the lackluster challengers, Cornyn spent $4.5 million in a four-month span ahead of his primary. Graham has spent at least $5.3 million in a five-month period leading up to Tuesday’s vote.

Cornyn defeated Stockman by more than 40 percentage points, and Graham is ahead of his six challengers by a similar margin, on the cusp of the 50-percent threshold he needs to avoid a runoff.

Cochran, the former Appropriations Committee chairman and current ranking Republican on the Agriculture Committee, did virtually no fundraising in 2013. In the first nine months of 2013, he raised just $400,000 — an insufficient pace for a junior House member, let alone a powerful senator.

When McDaniel entered the race in the fall, Cochran barely had $800,000 left in his campaign account despite not facing a serious challenge since 1978. The Club for Growth, Senate Conservatives and other anti-establishment groups then poured $5 million into Mississippi, giving McDaniel a financial edge over the 36-year incumbent.

Cochran also failed at the other plank of “overwhelming force,” which requires constant presence back home at public town halls and working behind the scenes with local officials.

Alexander, 73, assiduously tends to the needs of other Tennessee politicians. He worked the Nashville room at the state Republican Party convention in December 2012 as if it were already his own election year.

Before the state’s top GOP officials — a full 20 months before his primary — Alexander announced his reelection bid and unveiled his top supporters, including the governor, the GOP leaders of the state legislature and every member of the state’s House delegation.

Simply put, there was no one of any heft that was left to challenge Alexander, whose two primary opponents have garnered little support so far.

“I haven’t changed my voting patterns, but I’ve stayed connected,” he said this week.

McConnell took a similar approach in Kentucky, winning an early endorsement from the tea party’s first hero, Sen. Rand Paul (R-Ky.), and the 72-year-old then worked his state as if he were running his first Senate race in 1984. Paul is a frequent presence at his side.

For Cochran, no such public rollout accompanied his reelection announcement.

In fact, many supporters privately thought he was leaning toward retiring. Instead, in an early December interview with a Mississippi political columnist, Cochran announced he would run without any fanfare or show of support from other Republicans.

His public appearances were then few and far between, but when he did show up there were some stumbles, such as his admission of how little he knew about the tea party. He refused to debate McDaniel, and in the final days leading up to Tuesday’s vote, he had several unsteady performances.

In between campaign stops last week, he suggested to The Washington Post that he had to be talked into the race by a few close advisers. “I thought I’d served long enough,” he said.

Other Republicans have taken note.

Sen. Richard C. Shelby (R-Ala.), a 28-year veteran, expects to run for a sixth term in 2016 despite his own long record of bipartisan spending on the Appropriations Committee. Two years before he would face a tea party challenge, Shelby has a nearly $18 million campaign war chest and has already visited each of Alabama’s 67 counties this year.

“You need to work hard, you need to be heavily armed and you need to take nothing for granted. If you do that, things work out,” Shelby, 80, said.